How are LEGO bricks categorized? What are the names and categories of LEGO bricks? Is there a consistent naming scheme? Does Lego have an official name or identification for different pieces?
It's been worked out by multiple groups who decided on different systems. The disagreement is based on what information was publicly available from the Lego Group at the time, what pieces had been manufactured up to that point, and what features the category authors thought were salient.

To really appreciate the magnitude of the problem, just try classifying every part in your own collection. Make your own categories or try to follow somebody's elses. You will raise a lot of questions about how broad the categories should be. Everybody with a storage system goes through this.

Existing Classification Schemes


Consider BrickLink's categories, which are up to 188 categories containing 25,096 parts as of this writing. In this scheme, there are no subcategories, only 2 levels. The category Brick covers 70 parts, but has 7 co-categories like Brick, Decorated (1356) and Brick, Round (38). Brick, Round shares some close neighbors in a separate group, Cylinder(47) (both terms inherited from LDRAW, by the way.) You could reasonably argue about membership characteristics for classifying under Brick, Round or Cylinder or even whether there should be two groups.
LDRAW, originated by James Jessiman), didn't categorize parts hierarchically, but for many parts the first word or two makes a genus.  Programs such as http://mlcad.lm-software.com/ displayed some of these categories in a menu but invited you to think of the big list "like the rummage in the Lego box." As of this writing, LDRAW covers 5,377 parts.
Peeron is an example of an exhaustive set database using LDRAW's part numbers and names. Peeron didn't introduce a hierarchy of its own.
PartsRef, by Steve Bliss, a major LDRAW parts author, attempted to categorize the LDRAW parts hierarchically. It had 22 top level categories, for instance: Architectural, having subcategories like Doors. However the Plate category remained undivided and included all the plates with various protrusions. 
LEGO Digital Designer is a newer example with its brick palettes (sorry, I'm not a user.)
Pick-a-Brick as of this writing uses 42 categories, some of which are very broad. For instance, "Brick, Special" is one category (currently 155 parts) that other schemes divide into various groups for round bricks, modified bricks, cylinders, hinges, cones, etc.
Auczilla by Todd Lehman and Suzanne Rich (circa 1996-2000) had another exhaustive parts scheme that excelled in its use of geometric adjectives for part names, such as Half-Octagonal (debatable on math grounds) and Quarter-Round, Convex and Concave. The link is from Internet Wayback but shows the top categories. 
Finally, Technica (1999) was another attempt at classifying just Technic parts.


In the end, some parts are placed into categories primarily by geometry, others by function, yet others by historical origin. You could also conceivably group bricks by building function (I'm thinking of direction-changing rectilinear bricks) or even by simply the number of studs (I'm thinking of a Kanji dictionary.)

Part Names Make Classification Muddier

Consider the language barrier: if you read Lego's English Pick-A-Brick names, you realize it's translated (from Danish). A major category is "Brick, Bow"  also known as an Arch.

Dome



Consider 30367, LEGO's "Final Brick" filed under Bricks, Special. Perhaps they meant Finial? LDRAW calls it a Cylinder with Dome Top. BrickLink puts it under Brick, Round (remember BrickLink has both categories Cylinder and Brick, Round from its LDRAW heritage.)

Jumper Plate



Jumper Plate is one name that made perfect sense to Europeans but baffled Americans; to James Jessiman (Australian) it was a Plate 1 x 2 with 1 Stud. I recall Auczilla used the adjective Offset in there. In Pick-A-Brick it's now a Plate 1x2 With 1 Knob. Or should it be a tile because it's mostly flat?

Loudhailer



Then there are multi-purpose parts: describe 4349 geometrically and it's another cone. But it could be a Minifig Weapon, or just Minifig Utensil. In LDRAW, it's a Loudhailer and has no category as such. The American term would be a Megaphone but many know it as Star Wars Blaster. 

Multiply this complexity by thousands, and you get an idea that the task of naming is never settled. 

Part Numbers

Many LEGO bricks have part numbers imprinted from the mold. This is now called a "Design ID" by LEGO. It served as the original basis for LDRAW part numbers. However, LDRAW invented its own numbers for parts with no number. 

About the year 2000, LEGO introduced a 7 digit "Element ID" to designate a particular design in a particular color. These numbers began to be used internally at LEGO for Building Instructions software, and externally in LEGO Creator software products. They are now shown in official set inventories and in Pick-A-Brick.

Conclusion

This should give you an idea why different name and category schemes don't align. We start by talking about the same LEGO pieces but we bring preferred names from childhood or later (I always referred and still refer to 1xN bricks as 1-bumper, 2-bumper, etc.) We group them according to geometry or function, but find new reasons to subdivide or revise to make the groups more intuitive. New parts challenge these categories.  LEGO does not seem to have given the problem much thought or care, lumping parts into catch-all categories. I derive immense satisfaction from solving a problem of devising a part's genus and differentia, but I realize that just like in the taxonomy of living things, there is competition between systems, and endless revision.