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What to do about top students making other students lose confidence?
About once a year, I end up with a section with a student who is well ahead of others in the course. Often, these are students auditing the course, but sometimes are senior students attending 100-level courses or students majoring in the subject attending a course for non-majors in a made-simpler-for-non-majors course.
Such students are often eager to learn, quick to volunteer to come to the board or volunteer their work for peers to check (e.g. in a writing course), active in discussions, and they tend to take charge of group work. This does not seem like behavior I should discourage. Yet, in such sections, I notice significantly reduced confidence, participation, and engagement among other students, who seem uncomfortable with having a strong student constantly outdo them. Other sections tend to have active lessons, with many students eager to volunteer or join in a discussion, but in these sections, the class atmosphere is quiet and I'm reduced to calling on names and getting unwilling participants to carry the course along.
What can I do to turn this situation around?

I think it varies a lot by level.  I see this behavior a lot for undergrads.  In graduate school, I've seen people rise to the challenge of a stronger student.

Wow, does someone care for their students!

What to do when top athletes make others lose confidence? What to do when top musicians make others lose confidence?  That said, why require students to participate verbally? Some kids don't want to stand out regardless of their skill level.

It is quite important that students speak in the lessons. In this way, I can understand where everyone is in the material, what areas need more explanations, and what material I can go through quickly. The students also benefit from analyzing the thinking of their classmates or from hearing repeated information being clarified. If the lecture is merely a dialogue between me and a single student, other students start to fall behind quickly. Students who sit silently at the back of the classroom are often the ones who do poorest on the exams.

@CarlWitthoft Students are not in direct competition with one another. Everyone can succeed in learning the material and earning a passing grade. Since there are no trophies or first chairs up for grabs, I don't see any similarities between the OP's question and your sardonic questions.

@Jeff  and yet,  it's possible for JVs to enjoy a sport, and for second-tier musicians to enjoy playing in a group. My point is that there's more to both teaching and learning than accepting that you're in competition with the top-level kids.  Clearly the OP fears that the psychological equivalent of first chair is affecting the rank&file in his class.

@Village I don't really have an answer, but a suggestion. Don't rely on in-class dialogue to gauge student understanding. We have better tools now, and many of them are free (e.g. clicker systems, smartphone enabled interactive presentation software, even just paper cards the students hold up for multiple choice). Dialogue only samples from the talkative students anyway, which is not representative of the class as a whole. Using a better systen also alleviates the problem with the strong student: you will not need to call on them, because you ordinarily won't need to call on anyone directly.

@Village but if you look at the courses the OP is talking about it is like a semi pro athlete playing in a rec league. It ruins it for everyone else.

When I was a TA, I always found these types of students the most difficult to handle.  You don't want to crush their enthusiasm, but you also don't want to let them dominate and make it harder for other students to learn. It's also important to remember that just being way out ahead of other students doesn't cause this phenomenon: you're dealing with somebody who is both ahead and feels a need either to show off or to receive affirmation from you the instructor.
Some tactics that I found effective were:

Establishing clear ground rules that people had to raise their hands to be acknowledged, rather than just shouting out.  Then you need to become comfortable waiting long enough for other students to raise their hands too.
Saying things like "Everybody needs a chance to learn, so I'm going to make sure that we get some folks up to the board who haven't been there very often."
Rather than asking for volunteers, actively calling on students who have been silent.  It's embarrassing for them, but if you do it kindly and make it into a guided learning experience at the board rather than a test of their abilities, it can be a very good thing.
Privately discussing with the enthusiastic student, something like: "I'm very pleased at how well you're mastering the material, and I need to make sure other students have a chance to have the same learning opportunities, so I'm going to call on you less often."
Giving "extra credit" work that the enthusiastic student can be working on to occupy themselves.

In all cases, what you want to emphasize is the importance of active learning by participation, and how it's important for all students to have those opportunities.
Addendum: I was talking about this with my wife and she told me about a tactic that she uses that I think is another excellent addition to the toolkit:

Divide the class up into rough geographic zones and call on zones round robin, e.g. "Left side answered the last question.  Let's get an answer from the middle now."

+1 for number 4.  Conspiring with the enthusiastic student in order to create a better learning experience for the rest will make him feel important. You can for example ask him to count to 10 in his head before raising his hand. Or you can even ask him to ask a clarifying question if he feels the rest of the group is stuck. Asking a good unsticking question can be a real challenge.

I like 1,2 and 5. I've been on both sides as a student ("top" and "bottom"). I can confidently say that 3 would really make me nervous. I had a class where the teacher called people up randomly and I was constantly uneasy, lacking the confidence to come in front of many people. If you must do this, at least make it very clear from the beginning of the course so students know what to expect, or even better make up a schedule so they can prepare. On the other side, 4 would really bum me out, and I would see it as a sort of punishment for trying to learn.

+1 for #4. I've had a professor tell me that before and it's not disappointing to hear - in fact, it's reassuring knowing that the professor appreciates the work and effort. I don't fully agree with #3 though. I don't think this tactic practically (and effectively) applies to enough situations to be a useful solution. Also, unless everyone is getting the extra credit opportunity, in its current wording #5 seems a bit like favoritism and even though the 'top' students deserve the opportunity, is unfair to other students if only offered to those who are already doing better than everyone else.

Careful on the "conspiring" angle. You want everything you say to that student to be acceptable to any other stakeholder. It may be better to be said without eavesdroppers, but it should be said as if the entire class were listening. I agree with concerns raised by @ChrisCirefice; see my answer.

I always have disliked #3. Either I know the answer, so it feels like a waste of time, or I don't, which means I feel embarrassed being up there. It doesn't help me learn either because all I can think is.. "Can I sit down now?"

I'll offer two ideas; perhaps you can use them both:

One thing you can do is augment your class with on-line discussions, as a form of blended learning.
One advantage of asynchronous on-line discussions is that it's not as easy for one person to dominate the conversation. Also, if everyone is required to contribute to the discussion, no one can sit back and let the guru do the heavy lifting for them.
If your class is too big for all the students to answer the same question, the class can be put into groups, and each group can have their own shot at analyzing and answering the question.

During class time, when I've had a "resident expert" answering a lot of my in-class discussion questions, I've often let that person answer a question or two at the beginning. If they try to keep answering questions, though, I'll sometimes say something along the lines of, "Wait, you've had a turn; we need to get some of your classmates into the discussion now, too." Said in a friendly, pleasant and encouraging tone, it hasn't seemed to alienate the smart guy, and other students pick up on the cue that it's time for them to get involved.
I've also had some after-class discussions with these hard-chargers, to let them know I appreciate their enthusiasm, but it's best for everyone if they don't overdo it. That's usually been well-received.

I remember being that smartarse, and yes, they can be looking for approval or to show off but it depends.
If they're first year undergrads near the start of the year they may not be generally good at a lot of things outside the course, it can be a case of "Finally! finally something I can understand and really be good at." And they may be revelling in that unique experience. I wouldn't be too tough on that.
Further into the course if they're far ahead I'd suggest giving them something tougher to think about. They've read through all the sections you're covering in their own time, they're listening attentively but they're thinking about how it links up to something 3 chapters ahead. This is not going to help the other members of the class when that person asks questions that leave them lost.
My suggestion: Quietly give them something tough to chew on. Really tough. They're probably completing the assignments with ease, give them a challenge. It doesn't have to get them marks. If they're looking for approval or confirmation that they know what they're doing that will give it to them and it has the advantage that it keeps them advancing. If they're putting off other students actually talk to them quietly after class and tell them something similar to what you just said in this stackoverflow question and ask them to consciously avoid things which intimidate other students. They're not babies, they can understand.
Hell, give them a job tutoring undergrads in lower years.

Ideally, such a student wouldn't be in your course, they would be in an honors section or a senior/graduate level course that meets the requirements they are attempting to fulfil with your course while still being challenged. If you have an opportunity to identify such a student before the end of the add/drop period, you may recommend they make a change, or work with their advisor to give them an honors section while limiting their participation (so that others can contribute). I did such an honors course while sitting in on stadium hall sized lectures in my first year physics course at FSU, and my extra work consisted mostly of writing an on-topic paper graded by the same professor.
If it is too late for them to switch to a more challenging course, there's not much you can do, except to act fairly towards all of the students. You'll need to work harder at getting the others to participate. You could also give this student the attention they need outside of lectures, perhaps suggest to show up for your office hours, or speak to them before or after class.
Perhaps you could give the class, as an optional assignment, an opportunity to do a minor literature review of seminal works in the field, and you might communicate to your special student your expectations that they would want to do this. However, you can't force the student to do the optional assignment. But another possible upside for the optional assignment is that some of the others might surprise you with their self-motivation and thus build more confidence in their own abilities.

+1 for "give this student the attention outside of lectures" and "optional assignment". A lecturer who is interested enough to look for a solution that benefits everybody should definitely consider this option.

I was Adjunct Lecturer for some time in a technical field/institution and yes, I came across such type of students and situations.
What I've found out was that not strong students were feared to answer because they maybe say something "stupid" or "wrong" or anything that could be accompanied by a synonym of the previous adjectives. And, just by "luck", in such classes there always was a really good student who always knew the right answer
What I did to overcome such situations was to organize the lecture or the lab in order to actually looking for wrong answers in order to, on one hand, prove my point, and in other, motivate and make students to think about why I actually saying what I was saying in the lecture, i.e. what problem was solved with what they were going to learn in the lecture. So, if I was to point out the problem then I needed the problem and the problem was found in the wrong answers.
Thus, if the good student yield out the correct answer I accepted it. Actually, I was saying that this was the correct answer. But, just afterwards I asked "Although that this is correct, why is correct? What someone else would do? Why is wrong something else?" and trying to take answers from students that were not active in the lecture/lab. If the non-active students were not saying anything, then I asked "Why you do not say just what it comes to your mind", which most probably followed by "Because I do not want to say anything wrong". Such answer gave me the opportunity to "change the game" by pointing out really hard that "everyone is born without knowing and there are not wrong/stupid/etc answers from people that learning". When (finally) I got my wrong answer, I tried to: a) justify the student that gave that answer (because was a common answer, or an answer that was first came to mind), and b) start revealing the flaws of the wrong approach (and provide the path to the correct answer) with consecutive simple questions.
Although that such a strategy may took away some valuable time from a lecture, dialogs like the above were only held once or twice in the semester. Afterwards, most people were active and the good students were trying to actually go "deeper" in the problem than before. But, because "going deeper" required to find out what were the wrong approaches their "learning difference" from other students was diminished. This diminishing was happening because good students were actually waiting to listen my counter argument in the not-so-correct answers of their colleagues. By doing so (i.e. waiting for my counter argument), they do not yield out correct answers, they do not discouraging their colleagues and they learned to think one step ahead.

Students learn in different ways. Some like to talk out an explanation or answer and others like to reflectively think through their understanding. Sharing in a class of your peers (people who you want to think positively about you) can be difficult for many students, especially in a class larger than 20. One approach that has been documented as helpful for many students (and I use it as a staple approach) is the Think/Pair/Share structure. Ask the question, then have students write down on paper their ideas/thoughts/explanations. After a quiet minute, invite them to turn to someone nearby and together construct a stronger answer. After the students talk in pairs for 2-3 minutes, invite a couple of pairs to share their collective answer with the whole class. This is a structure with increasing risk (share with self, share with one other, then pair shares with whole class). I have found that after I carefully require the structure (e.g. keep everyone quiet during the reflective minute) several times, the students can execute the structure well and more students participate. I also am careful to explain why I want to use this structure often, rather than open questioning.

I think that the biggest detriment to isolating or adopting teaching policies that ostracize the enthusiastic student is that it demonstrates to the quieter students that enthusiasm is discouraged. The best case scenario is that instead of quieting down the top performer, is to discover additional top performers. That way, it's not just one or two very enthusiastic students, but a whole majority. That way, it becomes easier to single out quieter students and approach them outside of class.
From my own experience, I was in awe of a classmate who knew the course and the text very well. It made me look at my study habits and try new ones. I figure, if he wasn't as active, or if he kept his abilities to himself, I would have been less enthusiastic with my studies.
And, as a matter of definition or etymology the definition of student (technically study, however the definition of student refers to the definition of study) is

early 12c., "to strive toward, devote oneself to, cultivate" (translating Latin >occupatur), from Old French estudiier "to study, apply oneself, show zeal for; examine" (13c., Modern French étudier), from Medieval Latin studiare, from Latin studium "study, application," originally "eagerness," from studere "to be diligent" ("to be pressing forward"), from PIE *(s)teu- (1) "to push, stick, knock, beat" (see steep (adj.)).
Martha swanc and becarcade to geforðigene þan Hælende and his þeowen þa lichamlice behefðen. Seo studdede emb þa uterlice þing. [Homily for the Feast of the Virgin Mary, c.1125]
From c.1300 as "apply oneself to the acquisition of learning, pursue a formal course of study," also "read a book or writings intently or meditatively." From mid-14c. as "reflect, muse, think, ponder." Meaning "regard attentively" is from 1660s.

(All bold is emphasis on my part)
http://etymonline.com/index.php?term=study

I am not an educator, so please take these suggestions with a grain of salt:
I think existing answers are off the mark, as they're focusing on altering the environment in a very artificial way for the quieter students.  College is about learning life skills (not just academic ones) that will allow students to thrive in the real world, and they will need to deal with groups/teams of varying skill levels eventually.  Meanwhile, existing answers all focus on punishing (in some way) the most successful students.
Instead I would focus on the root problem.  There will always be someone better than us, and someone worse than us.  We should see the latter as an opportunity to help another while taking pride in our own skills, and we should see the former as an opportunity to challenge ourselves against a superior opponent, all within a safe and comfortable atmosphere.  Look for ways to encourage this sort of approach in the quieter students.
Specific suggestions:

Perhaps you could do more group work, pairing stronger students with weaker ones so that the former can help the latter.  Perhaps even formalize it with a 'peer mentoring' program.
Observe interactions and insure stronger students are getting and using the opportunity to increase their own confidence by helping struggling peers, not by overpowering them.
Give all experts the experience of failing safely, even yourself.  Creating an accepting environment for mistakes can help reduce the fear of making them; the classroom should accept excellence as well as those at a lower standing, without it impacting their valuation of self.
If you have the time, find specific skills that the weaker students are strong in, then give them the chance to shine in front of the class; use this to build confidence; consider making a point that everyone is strong in some areas and weak in others, a valuable life lesson.
As an extension of the former, and alternative to the first, find pairings that allow each student to use their strengths to shore up the weaknesses of the other, and assign group work based on those pairings allowing both to shine.

Maybe it depends on the institution, but the first, fourth and fifth suggestion seem impractical for a 100-level class as mentioned in the original post, since these tend to be pretty large at most schools. Even if the professor had time to get to know every student per term, there's also the issue of finding out what they're individually strong/weak in. I can also see people resenting a system as presented in the first  suggestion--getting pigeonholed as a "weaker student" early on and being assigned a "stronger" peer certainly wouldn't help anyone's confidence or desire to learn.

@MiloPrice Good points.  I went to a small university so my classes were never larger than 30 students; I don't have much experience with larger classes.  Regarding your concerns about suggestion one you're right that it would need to be handled very carefully; merging it with suggestion five may be more practical.  In truth none of us are strong or weak, we're a mix of both depending on what topic is in question.  I think the best approach is to find some way to use that truth to build confidence for all students.

@MiloPrice As I think about it more, I'm not sure I agree with you on the demotivating in number one.  Many professions still work on a mentoring system; in fact it's been pushed more and more.  We are all beginners and weak in a subject at some point, and we all become experts in our chosen fields at some point.  And we definitely all learn at different rates.  By college age I think these students should be mature enough to understand and be comfortable with that, and if they're not that's a lesson well worth trying to teach.

I'm not sure how other answers here focus on "punishing" anyone; they seem to be about ways to achieve a more balanced in-class discussion forum.

College is about learning life skills — [citation needed]

@J.R. I guess my perception was that treating someone who is excelling in the same way as someone who is not is 'punishing', in that you're giving them a lower reward:productivity ratio (maybe even reward:work ratio) than others.  This discourages excelling.  You could argue that people should excel for themselves and not for external validation, but human history and economics (for instance, the existence of positional or veblen goods, or the concept of prestige) tells us that most people simply cannot (or will not) do this.

@Nicholas I guess I just feel like the dynamic would be different in a professional context, where the mentor would most likely be someone who had worked there longer, than in a college class where everyone would (presumably) have had the same prerequisites and be taking the class for the first time. As a volunteer mentor in my department at the university I'm at, I agree that it can definitely be useful, but it sounds like you're suggesting doing it within the same class, with mandatory pairings. I'm at a medium-sized school on the quarter system though so my perception could be different.

@MiloPrice You may be right.  I wasn't thinking of it like that.  I think I've always put less weight on age than on maturity/skill and that may be biasing me.  Some students may have issues with this.

@Nicholas - Just because one particular student understands the topic better and is more vocal about it during class doesn't mean that student is "excelling;" it depends on the nature of the in-class discussion. Suppose I'm trying to give everyone "equal time," and my questions are designed to help people think a problem through (think Socratic method here), but one savant keeps blurting out an answer before other students have time to work through the problem mentally. That's not excelling, that's distracting. My classroom is not a quiz bowl, and knowing when to speak IS a "life skill."

@JR I don't disagree.  I read the question differently, and assumed these students were acting with proper etiquette, raising their hands and awaiting invitation to speak.  If they really are just blurting out answers then it becomes a matter of politeness that they need to be instructed on.  I interpreted the question text as implying that these students were better versed in the subject material, so I answered with that as a presumption.

@Nicolas - Whether they are blurting out answers out-of-turn, or raising their hands skyward almost immediately after the questions are asked, the effect can be the same: dampening the motivation for other students to learn. Some will behave as though, "I can stop thinking now; this fellow next to me already knows the answer." In group work, this so-called leader often takes on too much responsibility to the detriment of his peers (they don't learn as much). That said, we agree that this situation can manifest itself in many ways, and different circumstances may call for different solutions.

College students realize that there will always be someone more knowledgeable out there. They know there are going to be people that are majors, retakers, old serious adults, or students with genius parents. If you want to get your other students to participate then just randomly call on people whether they raise their hands or not. Tell them to pick group leaders by rock paper scissors.

I have edited this answer because it was unnecessarily rude and disparaging toward "teachers" and other users who answered the question, in violation of site guidelines on expected behavior. Please familiarize yourself with these site guidelines.

While in academia (from bachelors to after the PhD) we most certainly do realize there is usually someone more knowledgeable out there, it still causes a lot of discouragement (see e.g. this question). And, while randomly calling on students will get them to participate, the OP is asking for advice to motivate the students to want to participate, and your suggestions are not very helpful there.

This answer makes close to no sense. Yes, there are people who are more talented than others in everything. What does that have to do with the question?

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How to handle students not returning borrowed equipment?
I frequently bring my own equipment to the classroom for student use. Some students forget my announcements to bring certain items. Some items are infrequently used or kept ready on standby when time permits or need to be carefully prepared beforehand. For these reasons, I provide my own equipment, rather than requiring students to bring them.
The problem is, the students are walking out the door with my stuff. By the end of the year, I must go out and buy everything again. Most items individually are cheap (~$1 each), so I do not think any administrator or security officer will pay attention when I complain, however, the quantities add up over time.
I am surprised to find such behavior among college freshmen. How should I deal with these thefts?

Do you have reason to believe they are doing it intentionally? If not, I wouldn't describe it as "theft", and in that case, it may suffice to have a routine at the end of class where you or some student monitor collects the equipment, checking against an inventory if necessary.

It is possible that students take the equipment by accident. I'm not sure why students would do that.

Absent mindedness? Mixing it up with their own notes, pencils, etc that they just throw in their backpacks? What kind of equipment are we talking about?

Beyond the obvious collecting a form of collateral which works for any item but requires more work from you, it might be helpful to know what the items are.  If we're talking about something small then maybe you affix a piece of cardboard to it so it's harder to shove into a backpack or pocket.

If it is something frequently required for class, why do you have to use your own (personal) items, rather than equipment owned by your employer?

My old mathematics teacher would take a shoe for every piece of equipment borrowed. This meant they could not walk out.

@Alizter: I'm reading this on a day when I met a barefooted guy in the lunchroom.

@Alizter I am usually barefoot at university. Can I give a sock instead?

In my college, professors required students to leave their student ID cards as collateral when borrowing equipment.

@DanielHatton: Fair enough, but apparently, the situation the OP finds themselves in may require a reevaluation of whether providing one's own items is still preferrable to the time and stress for buying through official channels.

Unfortunately I do not have enough rep to write a comment on Moriarty's answer, but I think that it is most likely because of absent-mindedness in the part of the student.
I wanted to comment the following idea if you're lending out something like pens:
My mother used to work with students who would need to come in, go through a long document and sign it before leaving. She would go through an enormous number of pens, as the students often just popped them back into their pockets by accident.
She came across the idea of flower pens (gluing a big fake fabric flower to the end of the pen) and implemented it on the pens in her office, and she said that suddenly her pens stopped disappearing - since they were now tricky for the students to mistake them for their own, or accidentally stuff them in their pockets.
Depending on equipment you could try other things (I imagine you could spray-paint some things in bright neon or shiny colours, attach big tags or tassles to the items) to make them stand out a little more.
(A sample pen: http://ideasforfrenchclass.blogspot.com/2010/09/flower-pens.html)

I add an assignment in the grading system with a single point: "returned all borrowed equipment."
I don't actually use it in the calculations, but seeing it in the gradebook (which students can access) encourages compliance.

Ps. This is for equipment they borrow for a few days, so my solution does not quite work for the OP which is concerning in-class use...

What exactly is this equipment (if you don't mind me asking)?

In my case, field equipment like audio data recorders, mikes, etc.

I can see two solutions here:
1. Deter students from borrowing or walking away with equipment
If safety is not a concern, you could require a shoe as a deposit. No student will get very far after class without realizing what they're missing. It will save you having to remind your students to return said items.
You could also take a tally of all loaned items and release the class only when all have been returned.
Another solution is for the replacement equipment to be clumsy, comical, or embarrassing. Giant calculators might get the job done, but they won't be inadvertently taken out of class and you might not see many repeat customers.
Unfortunately, any deterrent besides loudly pronouncing at the end of the class "please return anything you have borrowed!", or perhaps taking a tally, may be difficult to implement without giving a distinct "I don't trust you and am going to treat you all like children" impression.
2. Stop lending out equipment they are supposed to have.
A more cynical point of view is that university students should be intelligent, responsible, and well enough prepared to bring any equipment that is either listed as a course requirement to have, or is generally expected (e.g. a calculator, stationery).
Your students are probably less likely to prepare properly for class precisely because they know you have spare equipment. If they're supposed to have their own stuff, make it the students' own problem if they forgot. They will only learn if failure carries consequences. If you abolish your loaning policy, I suspect the fraction of students coming properly prepared will quickly, and significantly, improve.

A shoe, their ID card, or driver's license are all good collateral.

"A more cynical point of view" - how is what follows an even slightly cynical point of view?

@Bill, I like the ID options, as it means even if they do walk out with the equipment, you know who to contact!

@BrianS Withholding an ID card is illegal in many jurisdictions. Also, some universities have explicit rules against holding student cards as collateral like this. It's morally dodgy, and needlessly humiliating without being in any way funny. Claiming a shoe as a deposit is awesome, though.

@G. Bach I think that the view that students are less likely to prepare properly for class if the teacher has spare equipment (despite the obvious cumulative inconvenience) is cynical. Likely true, but cynical nonetheless.

@SáT How is taking an ID card humiliating? I agree that it's (at the least) poor practice because it's very easy for the student to forget he handed it in as collateral. If you mean that it's humiliating because the teacher is appearing untrusting with regards to something of inconsequential value, I would agree. At least taking a shoe conveys the distrust with good humour.

@SáT: Isn't the shoe much more humiliating? Students may have to expose their smelly feet or their perforated socks to at least who sits close to them, which can be quite embarrassing. Compare that to handing over a small card that probably no-one can get a glimpse of ...

@O.R.Mapper It's embarassing, sure, but not humiliating. Everybody gets a good laugh, and an anecdote to tell for years. Of course, this does require a good mood in the class to begin with, and the prof must truly be witty to pull this off without it turning sour (and must be quick to improvise if someone looks really uncomfortable). Taking the ID just makes the prof look like a policeman, or a used-car-salesman, without any chance of making it look better (and there are the legal aspects to it).

@SáT: It depends on the person whether someone can look back to an embarrassing situation and laugh about it or feel humiliated. I fail to see how handing over one's ID card in exchange for a borrowed object could even be embarrassing to start with, though.

Wait. I don't understand. Making a list of their names says "I don't trust you and I'm treating you like a child" but taking one of their frickin' shoes says... er, what, exactly?

@SáT there's absolutely nothing humiliating about collecting a pile of student IDs  or driver's licenses when you had out the items. It also makes it easy to figure out who owes you them back at the end of class. You can simply call names going down the pile until have all the pens (or whatever) back. Everyone saw the students come borrow the equipment at the beginning of class, and it's better than having stinky-footed freshman hobbling around the room or making them late to their next class while they put their shoes back on.

@DavidRicherby The very same, just with a little joviality (SáT says it well). Noting down people's names when they borrow things just says "now, make sure you this back". Taking their shoe as collateral says "please bring your own stuff next time". I admit would often be an inappropriate solution, but it would work.

I don't agree with shoe. It may distract them from learning and decrease their attention.

@SáT Hm. A community college I attended used student ids as collateral for using books that were on-reserve.  Keeping a driver's license as collateral for major things is standard in many areas.  In fact, I think of my id card as primarily a device that I can give to the university as collateral for something. :\

I don't think the students are stealing as much as forgetting to give back your supplies.  I'd suggest helping them remember by bringing a jar that each students puts a dollar in when the supplies are handed out.  In order to get their dollar back at the end of the class, they must return whatever they borrowed.
I'd also get a label maker, and put something like "If found, return to Prof -insert name and office here-."  Make sure its gaudy and hard to miss.  This will both discourage actual theft (which I do NOT think most of your students are doing), and provide bread-crumbs home if its found under a pizza box after 3 weeks (the more likely scenario).

The second one is actually very good advise.

Speaking from my experience as a student only.
The professors that were responsible for my courses as an engineering student varied in strictness, some were lending stuff to student who forgot to bring their own, some were very, very strict. The latter professors plainly stated that forgetting item x will make you unable to complete the laboration, and an uncompleted laboration meant failing the course until the next opportunity to do the laboration.
Guess what, almost no student forgot to bring item x. Not only is this the right way because you won't have to bring item x, it also prepares student for the real world, were there won't be a professor doing your work, where stupid calculationg´errors because of negligence are as bad as not understanding the problem at all, etc. This principle should be applied across the board imo.

The issue is that you give out equipment to your students, but they have little incentive to bring it back.
As said above, you can create an incentive by making their grade dependent on whether they return them. The drawback is that you will have to keep careful track of equipment given out.
You can minimize the bookkeeping somewhat by lending not on an ad hoc basis every class day, but only once at the beginning of the semester, and ask them to bring the items during or after the final. You can also print out little paper labels, and for every item lent to a student, write on the paper the item and name of the student. Keep the paper until item is returned. You can even make the students prepare these papers.
You could "lease" the items to the students - sell them at cost, and offer to buy back when they are done. That way, you are fully insured against lost items. However, because of the money exchange involved, I suspect school administration will take issue.
You can also use your attendance sheet: Require everyone to come to your desk and sign the sheet before leaving the class. Refuse to sign their row unless they produce all borrowed equipment, and consider them absent for that class if the items are not produced. The drawback is that end of class will become chaotic and take several minutes.
In my opinion, the best option is to not lend items at all. Simply tell the class that they will not be allowed to attend the class that day unless they bring the equipment. Emphasize that you intend to stringently enforce this rule. Everyone will always show up with the equipment.

Yes, it will cause chaos at the end of the class.  The first few times.  After that, the students know what to expect and will bring the borrowed equipment back to you with a minimum of fuss.

Make yourself a note to say something to your class, 10 minutes before the end of your part in it (lecture, etc). "Remember, I have brought these XXX's for your use in case you needed them, I really need them returned by the end of class."
Modify as needed (i.e. turn 'end of class' into 'now' if needed)

I know many professors who lend things to the students such as books, their thesis hard copies, compact disks, etc; and they have a very simple way to track all the students who have borrowed things from them.
Prepare one or two A4 size papers, write "Borrowers List" on top of it and every time you lend something to a student; ask him/her to write his name, the date he/she has borrowed it from you, the date he/she will give it back to you, his email and phone number and his signature.
After the student has returned the borrowed thing to the professor, the student and professor, both, can sign under this paper that the borrower has returned it.
This will have many benefits to the lender and the borrower, both.

The borrower will remember that he has signed his name when he has got something from the lender; so he will try to give back that thing to the owner as soon as possible to avoid any difficulties in future.
When the professor asks for a signature from the student, he shows that he cares for his owning and it is important for him that the things he has borrowed should be brought back to the owner. Consequently, the student will do all his/her best to return that thing to the owner.
The professor will have a list of his owning which are lent to the students, so he will never forget what is borrowed to whom. If the borrower does not return the thing, the professor can simply send him/her an email or text message to ask the student to return the thing he has borrowed.

For a book, sure. But for something worth a dollar or so?

That is a tricky situation. The overhead that would come from noting every student what he took and whether it was returned is annoying and unnecessary. I think the easiest way to handle this would be to point out to classes beforehand that stealing equipment is inappropriate (obviously) and that you would take measures to discourage such behavior. Those measures might include that the class, from which someone stole something, would be required from then on to supply its own equipment, i.e. would be banned from using yours. If students don't have the appropriate equipment ready, they can't participate in the exercises or get their grade diminished.
The point is to relocate the burden to the collective, instead of carrying it yourself. Of course, in the end you will probably still have to pay for the missing equipment, since it is too small to pursue justice to great lengths, but I hope you'll get at least some small satisfaction.

Use accountability. Create a form with 2 identical halves, listing the name of the borrower and what they borrowed. When borrowing the item, they get half of the form, and you get half of the form. When returning the item, they give you their half of the form back with it, and you destroy both copies. Anyone who walks out without returning the item, place your half for safe keeping. If they need to borrow again and you have something with their name on it, deny them. There is an option to hold them financially liable, but IANAL. Make the policy clear in the syllabus.
This is an Army solution to a much more serious equipment accountability issue than classroom supplies, but it works.

Duplicate forms for something worth a dollar?

It doesn't have to be duplicate. The student doesn't need a copy of the form, they can be told that it's their responsibility to remember what they borrowed.

Do present/absent check at the end of your class session.
When you lend something to your student, put a sign in front of his/her name in your student's names list. At the end of the class, when you read the student's names to see who is present and who is absent, when you see that s/he has a sign in front of his/her name; ask him/her to give that equipment back to you.
If he does not return your equipment, do not accept his presence; until he returns it to you.
Allocate a significant amount of your final assessment to the students presence and absence in your classes and this will require them to not only appear constantly in class sessions, but also be required to give your owning back to you.
Your key is their exam papers/assignments!
On top of your exam/assignment papers, write the things that the student is borrowing from you. For instance, somewhere under his name or student ID. Even you can prepare a white block in which you can write the borrowed: title and in front of it, you can always list the equipment you lend to your student.
At the end of the exam/class session, when s/he gives the exam papers/copies back to you; check whether s/he has borrowed something from you or not and immediately ask him/her to return the borrowed equipment.
If s/he is not returning the borrowed equipment, then you refuse to take his solved papers.

This is based on the assumption that it is accident, not deliberate theft. I think I may have one more of a certain type of USB cable than a remember buying, but if so have no idea where I got it from.
Get some luggage tags, and stick on each one a label saying something like "Property of Professor Village. If found, please return to office/building/campus."
Attach one to each item. Tie, duct tape, or glue as appropriate for that item.
Most students will remember to return the tagged items at the end of class. If not, next time they look through a backpack and notice the tag they will be reminded, and will also know how to return it.

Make them pay a dollar each time they borrow something and give them a token -- like a plastic poker chip -- with the equipment.  When they come back with the something and the chip, give them their dollar back.
Around these here parts, that's how they do it with beer mugs and such.  No bookkeeping required!
Make sure they know you're not offering to sell things for a dollar -- they must return it even if they're willing to not get their dollar back.

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What does authorship ordering really mean in field X?
In my field (theoretical computer science), authors of any paper are always listed alphabetically; our papers don't have "first authors".  (Well... hardly ever.)  In most other disciplines, at least within science and engineering, the ordering of authors is a signal about their relative contributions to the paper, with the first author indicating the most significant contributor.  Hiring and promotion committees do give extra weight to "first-author papers" (and sometimes have to be reminded that not all areas have them).  As an outsider, I find this practice confusing.
What does first authorship actually mean in your discipline? I understand vaguely that the first author is supposed to be the one who "did the most work", but what counts as "work" in this comparison?  Does "most" mean "more than all the other coauthors together" or just "more than any other coauthor"?  What happens when the comparison is unclear?   How often is "did the most work" the actual truth, versus a cover story for a more complex political decision?
I realize that the precise answer is different for every paper.  I'm looking for general guidelines for how an outsider (like me) should interpret first authorship in your field.  Pointers to guidelines from journals or professional societies would be especially helpful.
Please give only one answer per discipline.
Answer list:

Math and related fields with alphabetical ordering:

Pure Math
Applied Math
Computer Science (1) and (2), see also human-computer interaction
See also physics

Physics, engineering, and chemical engineering
Biology, medicine and related fields, where order matters

Biology
Chemistry
Cognitive neuroscience
Epidemiology
Medicine*
Microbiology
Psychology

Earth Sciences
Art History

It's a "community wiki"; no best answer should be chosen, given the nature of the question.

I am in the field of Occupational Therapy. Does anybody have an answer for this question in my field? I am a recent graduate. Thanks! Karina

You may want to look at Information-theoretic author order for a thorough treatment of this question.

I must say that most of the answers don't say what first-authorship really means, but rather what it ostensibly means, which is not the same thing.

I think engineering should be included in the list of 'where order matters'

Pure Mathematics: All authors are assumed to have contributed equally and are listed alphabetically.  The American Mathematical Society has put out a statement to this effect.

Yep. Theoretical computer science inherited this attitude from math.

+1 for the link.  I know this is the standard, but I didn't realize that the AMS had issued an official statement to that effect.

it would be nice if some body like the ACM wrote a similar statement, so that young cs theorists can point to it

Since theoretical computer science is a branch of mathematics, you can legitimately point young theorists to the AMS statement.    But I agree that SIGACT should post an official "same here".

Ah, this explains why Tarjan is always the last name on his papers. I just thought he was being modest.

All authors are assumed to have contributed equally –– I always thought Mathematicians make sensible assumptions, or even provable ones.

It seems to be the case in financial mathematics too (with very few exceptions). The last paragraph of that statement is very true. It is "unfair" in the sense that many (especially in industry) are not aware of this tradition and tend to attribute the work mostly to the first author...

@Walter: I can't imagine where you got that idea.  Mathematicians are in the business of deducing provable conclusions from given assumptions.  How sensible the assumptions are, especially when they have something to do with the real world, is another matter entirely.

Being a mathematician, I consider this an outright toxic practice. The idea that everyone contributed equally to a paper is non-sensical. At least in many branches of the applied sciences they are honest that the senior authors might never have read the paper.

Theoretical computer science did not always have this assumption.  Back in the 90s, it was order of contribution - or at least it was in my own, personal, rather limited experience.

@Ambicion - part of what makes this practice work is that senior people who might never have read the paper are not part of the author list.  The threshold for being an author for a mathematical paper is quite high; there are several papers to which I have made small contributions on which I am not an author.

I guess I'd also say it's not really that people believe the contributions were truly equal; rather, the authors reject the notion that their contributions can be strictly ordered.

@Walter I view "All authors are assumed mohave contributed equally" as analogous to "All men are created equal" or "Defendants in criminal cases are presumed innocent until proven guilty." Even though these assumptions and presumptions may be false, we agree to treat people as if they were true.

Cognitive neuroscience. The first author (most of the time a PhD student or a post-doc) is typically involved in designing the experiment, running it, analyzing data and writing up. The other authors are mostly involved in some but not all of these steps. They will usually help out but not do all of the work (e.g. they might show the first author how to do some analyses, or they might make many useful comments on a draft of the manuscript). In the institute where I'm studying (in The Netherlands), all the papers where I am first author will also be chapters in my thesis, and all the unpublished chapters in my thesis could potentially become papers where I am first author. Those where I am only partly involved will be a chapter in someone else's thesis, and I will not be first author.
The last author is as important as the first one. It is typically the supervisor, and ideally the supervisor is heavily involved. In labs that grow too big, a post-doc (once trained by the supervisor) might take over this role, but the last authorship still goes to the supervisor. It's like a brand name, it tells you whose lab the work is coming from. If you know a bit about the field, you will know the general ideas the paper will revolve around. If two supervisors are involved, they have to work out whose name will be last. I know of one situation where the two believed in different outcomes of the experiment, and decided beforehand that the person who turns out to be right will get last authorship. Most of the time, though, the decision is based on who did more supervision, which is ideally agreed on beforehand.

I think a common but unfortunate way to see the ordering in biomedical sciences is: First author = the person who did the most work, including writing the paper; Last author = the one who paid (and often, but not always, supervised the project to some degree) for the work by means of instrument, time and lab-space. Shared first-authorships happen on some journals that admit it, however not so common perhaps

This convention also applies to psychology more generally.

Computer Science. Computer science varies by field:

Theoretical computer science generally follows the same conventions as mathematics: the ordering of authors is alphabetical.  Cryptography follows the same conventions.
In programming languages, computer systems (e.g., operating systems, databases, computer security, etc.) and other applied fields, the order of the authors is significant.  The authors are often listed in order of decreasing contribution; faculty or senior folks are typically listed last.  The first author often has led the design, implementation, and experiments presented in the paper or has contributed the most to these elements. Other authors may have contributed more in total, or even individually, to these components, but sometimes at the direction of the lead author. The lead author may also have been considered to be primarily responsible for the writing of the paper, though not always.
In cases where the lead is shared between several people, papers can have multiple "first" authors, listed alphabetically, followed by an alphabetical listing of the other junior authors, followed by the senior authors. I have seen this fact listed explicitly on a CV. Usually, a PI comes last even if he or she provided the bulk of the leadership of the project; a PI coming first is indicative of an unusually high level of contribution from the PI / low level of contribution from the junior authors.
Overall, the meaning of first authorship ends up being vague enough that usually you have to explain the level of contribution explicitly in reference letters as such.
HCI (human-computer interaction) follows similar conventions to those in computer systems.  The authors are listed in decreasing order of contribution.  The first author is generally the person who both had the "main idea" and led the effort to ensure that the efforts to carry out the research and write the paper occurred properly. The authors are generally then decreasing in order of their contribution.

What does "theoretical computer science" exactly mean? I'm a PhD student in programming languages and even at POPL (our most theoretical conference) first authorship counts.

These observations follow my own experience. However, it should be noted that some professors (and, hence, their students and postdocs) use alphabetical ordering despite being in a generally 'in order of contribution' field. In my experience it's usually profs whose names start with [A-C] that seem to be keen on this :D anyway, the practical confusion is small, as those professors are well-known in my field.

Another comment: JeffE asked various times how 'most contribution' is defined if different people contributed to different aspects of the project. In computer systems / software engineering, this is mostly a non-issue. Basically all projects are concepted, executed, and written up by a single person (often a PhD student, and the first author). For the remaining authors, the ordering can get a bit muddy, but nobody really cares whether they are 2nd or 4th author.

@xLeitix: Umm, I wouldn't say that all projects are the work of one person. Not in Academia and especially not in industry.

Very late to the question, but the answer to Blaisorblade's query is that this is referring to combinatorics, algorithms, graph theory, complexity theory, computability theory, and related topics which are commonly collectively referred to as "theoretical computer science" or "theory of computing." This is distinct from "the most theoretical topics in each field across CS." Big-name conferences in this field include STOC, FOCS, and SODA

Medicine: The first author is the author. He or she is credited with the bulk of the work, and some even consider first authorship to be the only authorship of value. This may be partly due to the fact that a medicinal paper often has many authors, with some having done next to nothing for the paper (maybe read it). Although journals would like to discourage this, people write their colleagues' names on their papers, so maybe their colleagues will do the same for them and both get a more impressive publication record.
Sometimes you see asterisks above the first two authors' names, indicating that "both authors contributed equally", although it seems to me that this is in general not well recognized. I've been told journals want one main author. Also, for many academic positions a given number of publications is required, with some minimum number first authorships.

Biomedical engineering matches this description, too.

This is essentially also true for Epidemiology

I believe this applies, more or less, to all biomedical sciences

"with some having done next to nothing for the paper" - Eh, in my experience often these limited contributions are from other physicians who enrolled patients. They often have had a medical role in the study even if they didn't have a design/analysis/writing role. Journals have begun to discourage this, though.

Applied mathematics.  The first author is usually the one who contributed the most.  However, sometimes the pure mathematics convention of alphabetical ordering is used; this may be expressly declared in a footnote.  There are no official guidelines from SIAM.
There is no significance to being the last author, and only those who contribute substantially are listed as authors.  If not directly involved in performing the research and writing the paper, the supervisor is typically only listed in the acknowledgments.

Again, what does "contributed the most" actually mean?  The most text?  The most code?  The most ideas?  The most proofs?  The most analysis?  The most graphs?  All of the above?

@JeffE there is no generally agreed-upon metric.

@DavidKetcheson obviously describes an 'applied maths' community completely different from mine - I'd be hard pressed to name a single paper in my version of 'applied maths' that doesn't stick to alphabetical order. His 'usually' certainly doesn't apply to the applied-math community that I'm part of. And we never discuss that in a footnote. 'Applied maths' seems to be a big place ...

@MarkPeletier I am very curious, so I looked at publications listed on your webpage.  They seem to strongly contradict your comment -- for instance, on http://www.win.tue.nl/~mpeletie/Research/PubsElastics.shtml, every multi-author paper violates alphabetical ordering.  I'm not saying you're wrong -- I haven't looked at all of your publications, by any means.  But I think if you leaf through most SIAM journal issues, you'll see lots of non-alphabetical orderings too.

@DavidKetcheson Interesting remark. You made me look into my own publication list :-)
Considering that page: there are only two multi-author papers on that page, but you're right, both violate the alphabetical ordering. But in both cases, there was debate about this. The seven-author paper was published with some engineers, and the mixture between cultures led to non-alphabetical ordering. In the other case, one co-author also felt that alphabetical ordering would be wrong, and although I disagreed, we ended up changing the order.

@DavidKetcheson Looking through my own list of publications, I now see the general pattern: much of my work is with non-mathematicians, even when published in SIAM journals, for instance (since you mention those). In such collaborations alphabetical order usually doesn't win (despite my attempts :-). But when publishing only with (applied) mathematicians, the order is invariably alphabetical.

@DavidKetcheson: apparently your applied math community has nothing to do with mine. In my community, only person does the entire work and everyone else just gets dragged along. In fact, I'm afraid to say that your answer is probably just completely wrong.

I have stated an opposing view. No personal offense, but I consider your statement completely wrong.

Engineering: A first author is usually the lead student or worker on the particular project from which the paper originates. If there are multiple people working on a common project, then the authorship goes to the person whose results are most prominently featured, and who has done the most work in preparing the manuscript for publication.
A significant exception might be in multi-part papers, in which the first authorships may be shared among different people to recognize equality of contributions throughout the combined work.
The last author is often a professor, who advised or directed the lead author, but may have done little work on the project themselves.

But what does "lead student" mean in this context?  Specifically, what happens if the student whose work is most prominent and the student who did the most writing are different?

Or does that really never happen?

This has happened twice in my career. In one case, the PI was the lead in the work, but I was the primary writer. So he took the last position and offered me the first position. In another, I was the main writer, but not the primary worker; in this case, I ended up second on the authorship list. But, in general, I think this is a relatively rare event, particularly at the graduate school level. (This may occur more often when everybody is "staff-level.")

"first authorships may be shared among different people", how this may happen when we see the names appear in order, I mean, one name comes sooner, then the second, then the third… I have never seen two names as first authors on the papers, for instance.

@enthu While still relatively rare in engineering, in biomedical fields it is done with a special designation in the author list. In some areas, it is so common that it is included in the journal submission forms, and the designations are applied automatically by the system.

@enthu Similar to other fields, you designate the first and second authors with a "These authors contributed equally" footnote, and both parties can claim it in their CV as a co-first author. See this question.

Chemistry: similar to cognitive neuroscience and microbiology and many other fields - the first author is usually the individual who put most of the labor into the work. The PI, usually the last author, may have come up with the idea, but the first author usually does most of the following work: designing the experiments, synthesizing and purifying the compounds, collecting and analyzing the data, and writing the paper. The other authors might be: A student in a collaborating group that conducts an important, specialized experiment for the first author, a junior student in group who prepared some of the intermediates and collected routine data to help the first author and to learn the workings of the group, or a consulting professor offering expertise in an area that the first author and PI are weak in. The last author is usually the PI.
Some journals are beginning to ask for specific descriptions of the contributions of each author to combat vanity authorships. You also occasionally see the note that two or more authors may have contributed equally, but some journals discourage this practice also. As an example of this sort of declaration, the following statement was attached to the final article from my thesis:

Author Contributions
B.N.N. and S.Z. were coequal in their contributions and should both be considered first authors. B.N.N., T.Y.M., and G.R.H. proposed the project and designed the experiments. B.N.N., S.Z., J.T.A., and P.C.M. performed the synthesis and characterization. C.M.C. and G.R.H carried out the calculations. B.N.N., S.Z., T.Y.M., and G.R.H. assembled the data and wrote the manuscript.

Historically, the practice of putting the PI last is relatively recent. The PI used to be listed first, so that the PI was easier to identify and the collective works of the PI were easier to find in printed catalogs systems (where articles wee often indexed by the first listed author only). The historic order would have thus been: PI, first author, second author, etc. The advent of electronic databases removes the need for the PI to go first, though there are some who still do it that way.

"Coequal"???????

Biology: I felt the answers for medicine, microbiology, and epidemiology may not give the complete picture. Of course this is my own opinion, as there are no real formal rules.
The unofficial rules: In biology, the first author is the person whose contribution is larger than that of any other author. It is cannot be the author that contributed more than the combined contribution of all other authors - this definition doesn't even work mathematically (a 25%/35%/40% contribution paper would have no first author).
However, the situation is more complicated. Usually, the subsequent order of authors is according to decreasing contribution. Also, at the end of the authors list the scenario is mirrored: The last author is the senior author (i.e. the PI) that contributed the most, with the order of senior authors again reflecting their contribution (mirrored).
Then, it gets even more complicated. In some cases, you can have co-first authors. This is usually marked by the journal, indicating that these authors had equal contribution. Then, there is the "corresponding author" mark. Some (but this is less widely accepted) use this is to signify equal contribution of the senior authors, so for example you would mark both last 2 authors as "corresponding authors".
Practical issues: While it may seem to be silly to people not used to this method, the order of authors is actually quite important. For graduate students and postdocs, fellowships and prizes will often only consider your first-author papers as your "real" papers - this is usually written in the rules (you may asked to list only first-author papers). Furthermore, if you are co-first (equal contribution), you will often be required to detail your exact contribution (sometimes your supervisor needs to detail it as well in these cases). For PIs, the situation is similar - funding agencies will often only consider your last author papers.
Another less important issue is association with the paper. A paper will be generally referred to by the first author's name, e.g. "Smith et al.". If you are the first author you will be immediately associated with the paper. If your paper is high-impact, there can be benefits to this in terms of establishing your name in the field. This is one reason why even "equal contribution" may not be considered really equal by some.
Biology vs. other disciplines: Finally, I want to explain why this practice might be useful in biology and how it is different than math or CS, for example. First, any graduate student or postdoc is always under supervision. It is customary that regardless of the actual amount of involvement of the supervisor in a project, the supervisor is always listed as the senior author. You have to remember that it is quite rare for PIs in experimental labs to actually do any actual work themselves (this is different from theorists). This is not to say they cannot be highly involved. Then, many projects are collaborations between multiple research groups. It is very common to see 15-20 authors on a paper, and recently there have been many papers published by research consortiums, having hundreds of authors (although in that case the order of the author list is slightly different). One author could really be doing much more work that some other author, which is on the paper just because he/she contributed some biological sample or ran some program.
Is it good? I don't think this system is optimal. It can lead to personal conflicts and affect people's careers. Some journals try to bypass this system by adding a section detailing the individual contribution of each author, but this isn't widely recognized. Some funding agencies ask you to quantitatively mark the contribution in percentage of each author - but how do you do that? It is extremely difficult to quantify. One author spent a lot of time doing experiments, and another spent a lot of time analyzing the results - who should be first? It is very subjective and in the end is often settled by politics.

Replace "bypass this system" by "move this system to someplace else in the paper"...

@einpoklum I think they can potentially be more helpful, since they give more details on what each person did, and do not define how important each part is. Ideally, readers could decide for themselves how to weigh the importance of each part.

What you suggest is actually what is being done in these "Contribution" sections. The only problem is that the journals still keep the authorship order and that is the main thing people look at. The problem is not the journals, but that the whole system works this way - most evaluations (career, funding) attribute high importance to this order.

So people need to have the courage to be the first to switch to alphabetical, and indicate that is the order they're using. I know it's easy for me to preach, but it's still what I think could be the way.

@einpoklum it has to start from the top (funding agencies, academic institution policies, top scientists), I think. If someone at the early stages of their career say they don't care about author order, most chances they just won't have a career.

Epidemiology: First author generally means the author who did the bulk of the writing, and is likely directly responsible for the analysis of the data. The last author is (often but not always) the project's PI, a senior member if its a multi-site collaboration, or a place where someone who contributed heavily in some aspect, but not as much as the first author, goes.
Generally speaking, first authorship is considered the most important, last authorship has some benefit in terms of establishing a mentor role or the concept of the author as a senior researcher.
The exception for this is a small number of "pairs" of methodologists who tend to write papers together, which end up getting seen as a kind of equal contributors.

Microbiology: Similar to cognitive neuroscience: Ph.D student is first author by virtue of having done most of the work, and the PI is the last author. If it is agreed upon that more than one person did "first-author-level" work, then the authors are listed alphabetically with a footnote noting this fact on the title page.

Does every paper in microbiology really have most of the work done by a PhD student?

No, certainly not! The first author is often a post-doc (if they are the person who did the bulk of the work). As they move up the food chain, they might have several PhD students/assistants/other students working under them who do the hard work in the laboratory under their guidance - then they would typically move towards a more senior author position. Occasionally a PhD student or post-doc does 90% of the work and leaves science without writing up their research - then a lucky student or new postdoc may get to write it up and claim pole position. There's diversity out there...

Chemical Engineering: The first author is generally considered as the main contributor. In case there are multiple students who made equal contributions, then this is specified as such in the list of authors. (Mostly by adding an asterisk on the names and a footnote explaining the asterisk). Certain groups follow a policy of Adviser first and then rest, though it is considered as arrogant (This is prevalent in mostly Chemistry related sub fields).
In Machine Learning/Applied Computer Science the policy is again similar as Chemical Engineering with Student first and Adviser last, if there are multiple advisers then the advisers tend to rotate between different papers from the same project.
Finally in Medicine especially in General Medical journals, There is a detailed statement of contributions.

E.g.  Author Contributions: Dr De Wals had full access to all of the
data in the study and takes responsibility for the integrity of the
data and the accuracy of the data analysis.
Study concept and design:
De Wals, Deceuninck, Toth, Boulianne, Landry, De Serres. Acquisition
of data: Deceuninck, Toth, Boulianne.
Analysis and interpretation of
data: De Wals, Deceuninck, Brunet, Boucher, De Serres.
Drafting of the manuscript: De Wals, Deceuninck.
Critical revision of the manuscript for important intellectual content: De Wals,
Deceuninck, Toth, Boulianne, Brunet, Boucher, Landry, De Serres.
Statistical analysis:
Deceuninck. Obtained funding: De Wals, Boulianne, De Serres.
Administrative, technical, or material support: De Wals, Deceuninck,
Toth, Boulianne, Landry.
Study supervision: De Wals, De Serres.

Another important point is how are paper cited, from what I remember esp. in chemical engineering. A paper is generally mentioned as Last_name et al. and if there are only two authors or two equally contributing authors then it is mentioned as Last_name_1 & Last_name_2 et. al or just Last_name_1 & Last_name_2.

Just to clarify (I stumbled a bit): last name = surname, not about the order of the author list: Last_name_of_first_author et. al.

Computer Science This really depends on the institution and the group. In one group I've been working in the ordering was always alphabetical and doing otherwise would have been considered impolite. In another group, the PhD-first-boss-last principle was used.

And if there was more than one PhD student involved?  (Or for that matter, more than one "boss"?)

Then it would depend on their respective contributions, or it will be done alphabetically.

Then how do you tell whether a given paper's authors are alphabetical because they're equal, versus alphabetical because the alphabet happens to agree with contribution rank?

Ultimately, there is no way to know. It is part of the group culture, IMO and as such subject to multiple (mis)interpretations.

@AlexanderSerebrenik: Unfortunately? You mean, fortunately :-)

Earth Sciences (Physical Geography, Geology etc.). Authors are listed according to their intellectual contribution to a paper. The first author named on the paper is thus the person who has contributed intelectually the most to the paper. The second, third etc, names have decreasing importance (contribution). If more than one person can be  considered first author, those names are listed alphabetically and a note to the fact is made in the acknowledgement.
Only persons who have contributed intellectually to the paper are included. Lab assistants, techncians etc, are thus not included (although it still happens).
If a leading scientist, project leader etc. is not first author the lead role may be indicated by refering to that person as "Corresponding Author". This is common when first authors are junior contributors (students).

In the computer science/software engineering and human-computer interaction, the first author is generally the person who both had the "main idea" and led the effort to ensure that the efforts to carry out the research and write the paper occurred properly. The authors are generally then decreasing in order of their contribution. There is generally no consistent policy of putting the PI last in SE or HCI.
Thus, the policy is very similar to the Applied Mathematics answer posted above and quite different from the theoretical Computer Science areas.

But what does "order of contribution" mean?  How do you judge whether one coauthor contributed more or less than another?  And who comes first if Person X had the main idea, Person Y led the research effort, and Person Z wrote the paper?

In my experiences, the authors would agree upon themselves what the author order should be if the division of labor ended up being that diverse. However, in disciplines where author order matters, the question of who should be first author can be hotly debated (and is likely a good topic for a StackExchange question).

I merged this answer into the answer about Computer Science that was listed higher up.  (FYI, I disagree with the statement that in software engineering there is no pattern to put the faculty last; anecdotally, that's not consistent with what I've observed.)

who should be first author can be hotly debated (and is likely a good topic for a StackExchange question) — I agree!  But isn't that this StackExchange question?

I interpreted this question as "What does first author mean" as an end goal, rather than "How do we determine who the first author should be?" which I see as a process.

Astronomy: Generally alphabetical author lists are reserved for collaboration papers. The largest ones may even have the first author as the collaboration's name (See for instance this LIGO paper or this  arxiv version though the journal version ended up with Abbott et al just to confuse things).
For papers with smaller author lists, the order would be expected to be in order of contribution.

The First author will have done most of the work, written most the paper, and built the idea to start with. They are also usually the corresponding author.

Middle authors will usually be in order of contribution (sometimes its alphabetical depending on the size of the author list).

Last author Does not convey any explicit meaning.

Psychology: There are two distinct traditions to determining author-order in psychology, which we might label the APA and the biomedical traditions. The APA tradition orders authors by level of contribution. The biomedical approach places special importance on the last author, who is often the supervisor of the work. Finally, given the clash of these two traditions, a hybrid model is also emerging where first authors are most important, second and last authors are equal-next-most important, and then third through second last are in descending contribution.

The first author is usually the person that led the project, analyzed the data, and led drafting of the manuscript. They may or may not have collected the data personally.
The second author: The meaning depends on the author-order tradition applied. Under the APA tradition, they would typically be the second-most important contributor, and where the project is a student or post doc paper, they are often the supervisor. Alternatively, in the biomedical tradition, their contribution is typically not considered to be as important as the last author.
Other middle authors are usually involved in one part or another, lending statistical expertise, or subject-matter knowledge. They may also be junior people that collected the data (e.g. interviews, operating technology like EEG nets, etc.).
The role of last author(s) depends on the author order tradition implied. They may  be the least important contributor. Or they may be the supervisor or PI whose lab collected the data, obtained funding, etc. Data collection in psychology can be expensive and difficult, and take a long time. There can be multiple.
Corresponding author is determined by various rules and is given minimal significance with regards to contribution. Corresponding author is most commonly the first author. If it a student-led work, it is also common for it to be the supervisor. Finally, it might simply be someone with a permanent position who might be able to answer questions years later after the first author has moved on.
Shared first authorship is rare, but it is becoming more common. Alternating first authorship in close collaborations also does occur.

How do you determine whether last or second authorship implies a greater contribution in a psychology manuscript?
In general, there are a number of indicators that may suggest which author ordering principle was applied on a given paper.  Last authors are more commonly important (a) in fields aligned with medicine, neuroscience, and health, and (b) in papers with many authors (e.g., 6+). Second authors are more commonly important (a) in more  prototypical psychology subdisciplines like social psychology, personality, experimental psychology, and industrial-organisational psychology, and (b) in papers with fewer authors. In general, this seems to reflect patterns of collaboration. For instance, if academics are collaborating with the biomedical field, they absorb that tradition.
More generally, second and last authorship tend to be more important roles where the first author is a student or post-doc. In those cases, it is often the case that either the second or last author supervised the work.
Given the ambiguity in psychology about the relative importance of second and last authorship, some academics in psychology will highlight the number of times they have been a lead author, which is often specified as first, second, or last authorship.
APA Style Manual 7th Edition
The Style Manual has the following to say in Section 1.22
With regards to professional authors (e.g., where it is not the work of a doctoral student):

Principal authorship and the order of authorship credit should accurately reflect the relative contribution of persons involved... The general rule is that ... principal contributor appears first, with subsequent names appearing in order of decreasing contribution. In some cases, another principal contributor appears last. These conventions can vary from field to field and from journal to journal... If authors played equal roles in the research and publication of their study, they may wish to note this in the author note.

Art history: The humanities are still heavily invested in solo authorship. The first author is generally considered to be the one who researched and wrote the vast majority of the paper.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
How to proceed when PhD advisor leaves academia or moves to a new university midway?
How can a PhD student proceed when his/her advisor quits midway?

What should be the student's considerations when the professor offers him a choice to move along with him? Should it be the comparative rankings of the universities? Or the relationship enjoyed with the professor thus far? Or the progress of the work?
Do universities offer to waiver coursework and ensure faster-than-usual graduation for students accompanying new professors? Can papers published in the older university be considered a part of the thesis that will be written for the newer department?
What should a student do if such an option is not available to the professor and also there are not other professors in the department who could or are willing to guide the student? Will the university offer a compensation for him?

One issue is what your advisor is doing.  If your advisor is leaving academia (e.g., going to work on Wall Street or for the government), then you are probably on your own, but moving to another university is generally not a big problem.
If you haven't started doing research yet, you should probably just switch to a new advisor, but if you are already making progress, then staying with your current advisor is often the best approach.
One possibility that wasn't mentioned in your question is moving physically to the new university as a visiting student while staying enrolled at the old university.  You still work on research with your advisor, and at the end receive a degree from the university you started at.  Maybe it varies between fields, but this is pretty common in mathematics, and it avoids some of the difficulties like transferring credit or different requirements.
If your advisor is supportive (and they should be!), then this is generally not hard to arrange.  It's easiest if they are technically going on leave from the old university, rather than resigning immediately, but that's a pretty common way to arrange moving between universities, partly because it simplifies situations like this.  (And even if your advisor is not going on leave, you just need a colleague to step in as the formal advisor, while letting your advisor handle the day to day interaction.)  The major reason for difficulty would be if there was some serious problem at the old university, such as a personality conflict with the department chair, which could make the department unwilling to be flexible.

I went through this experience, and I have friends who've gone through this experience, and it's never fun. You ask a lot of questions; I'll try to answer as many as I can.

What should be the student's considerations when the professor offers him a choice to move along with him?

Consider the following:

How far along are you in your work? If you haven't proposed yet, it's probably easier just to find a new professor and start anew. If you haven't done any serious research yet (1-2 yrs), definitely find a new advisor and start fresh.
Did you have a good relationship with this person? Do you want to continue working with them?
Often, your credits will not follow you. (I don't have a source for this statement, other than I've been told by numerous people that graduate credits rarely transfer between institutions.) Make sure they will transfer, or that you will be given some sort of pass, before transferring.

Other stuff (rankings, location, collaborators) should obviously be taken into account as well. In my experience, most students do not move along with their advisor.

Do universities offer coursework waiver and faster-than-usual graduation for students accompanying new professors?

Almost certainly not.

Can papers published in the older university be considered a part of the thesis that will be written for the newer department?

Probably, talk with the university before transferring.

What should a student do if such an option is not available to the professor and also there are not other professors in the department who could/are willing to guide the student? Will the university offer a compensation for him?

That's pretty unusual. This happens all the time; people are familiar with the situation. In many cases, the department will be willing to help you find someone new. You should view the ordeal as identical to when you chose your initial advisor; you'll probably do (shortened) interviews with a few profs, talk to lab members, look into research, etc. The difference is that, by now, you should be familiar with those people who do research similar to what you've been doing, so your search will be easier; you'll know them, and they should know you, even if only because you've taken a class with them or something.
You will almost definitely not receive compensation. C'est la vie, my friend... welcome to the real world.

"Do universities offer coursework waiver and faster-than-usual graduation? - Almost certainly not". True in many cases, but "almost certain" is a bit strong. I know at least 2 professors who included these things in their hiring negotiations with the new university. Lesson to professors: you should feel morally obligated to at least ask the question of what will happen to your current students prior to accepting a job offer. Quite often these are very minor concession from the new university compared to other things you will negotiate. If you ask after accepting, then the answer is usually no.

I differ on the following: 1) I don't think whether one has proposed is a good indication of whether she should move. In some fields (e.g., economics or business), the formal proposal does not happen until the student is almost ready to defend. 2) In my institution (a major public university in U.S.), external credits up to 30 hours can be transferred into a PhD program. (to be continued)

(Cont'd) 3) My institution also offers the possibility of significantly faster-than-usual graduation for students accompanying newly-hired professors. For example, one student I know moved here with his adviser, and finished with his PhD in just two years - while the program typically takes at least six years to graduate.

I generally agree with @AnonymousMathematician that the easiest thing to do may simply be to visit the New University while staying an enrolled student at Old University. It may require checking with both departments, but then again so will and out and out transfer.
Another option that hasn't been mentioned:
It may be possible to convert your current advisor to an "outside reader" or simply a committee member who happens to have an appointment at another institution. The formal title of your "advisor" can then be switched to a member of your committee at your current institution - preferably one sympathetic and supportive of your situation.