What does it mean to comment on scripture with scripture? What does it mean to "comment on a verse with a verse" or "only scripture can explain scripture"? I see these kinds of comments popping up in comment threads that discuss the meaning of a verse.

What exactly does this mean, and how is it useful for resolving disputes? Can you cite examples of this where it makes the difference between two different understandings? Are there times when it is not the best or most relevant method? What is the alternative?
This concept is actually quite simple: other Bible verses are very useful in helping us figure out what a certain passage in the Bible means.  Since the Bible is one complete, coherent, interrelated revelation from God, we can use one part of the Bible to help us understand another.  There are several ways to do this:


Context, context, context! So often the chapter or chapters surrounding a passage will give all kinds of obvious clues as to what that passage means if we just read more and think about it.
Use clear passages to show us the meaning of obscure passages
When trying to understand the meaning of a certain (Greek or Hebrew) word, use other usages that use the very same word (or root word) in an obvious way
Use detailed passages to help explain vague passages
Use actual commentary found in the Bible to explain the original meaning.


(Perhaps there is more that I can't recall right now.  Can anyone think of ways I have missed?)

The last item actually can happen more than you think.  For example, Hebrews 11:17-19 is Paul's commentary; we should rely on it in order to properly understand Genesis 22.

We also have to be careful not to take this too far; for example, a certain word could have had a range of meaning even when used by the same author.  Thus we should not always assume that just because two passages use the same word, that word must necessarily mean the same in both.  The same word used elsewhere is a good clue, not proof-positive of meaning.  (Note that the "center column" bible references in nearly all study Bibles are almost always made of key passages that use the identical words).

Two good books that elaborate on this are Sproul's Knowing Scripture and Fee's How To Read the Bible For All It's Worth.