Am I the only person here who really hates them?
Generally, I don't use them unless I have to. I just really hate having to read something like
select a.id, a.region, a.firstname, a.blah, b.yadda, b.huminahumina, c.crap
from table toys as a
inner join prices as b on a.blah = b.yadda
inner join customers as c on c.crap = something else
etc
When I read SQL, I like to know exactly what I'm selecting when I read it; aliases actually confuse me more because I've got to slog through lines of columns before I actually get to the table name, which generally represents information about the data that the alias doesn't. Perhaps it's okay if you made the aliases, but I commonly read questions on StackOverflow with code that seems to use aliases for no good reason. (Additionally, sometimes, someone will create an alias in a statement and just not use it. Why?)
I think that table aliases are used so much because a lot of people are averse to typing. I don't think that's a good excuse, though. That excuse is the reason we end up with terrible variable naming, terrible function acronyms, bad code...I would take the time to type out the full name. I'm a quick typer, though, so maybe that has something to do with it. (Maybe in the future, when I've got carpal tunnel, I'll reconsider my opinion on aliases. :P ) I especially hate running across table aliases in PHP code, where I believe there's absolutely no reason to have to do that - you've only got to type it once!
I always use column qualifiers in my statements, but I'm not averse to typing a lot, so I will gladly type the full name multiple times. (Granted, I do abuse MySQL's tab completion.) Unless it's a situation where I have to use an alias (like some described in other answers), I find the extra layer of abstraction cumbersome and unnecessary.
Edit: (Over a year later) I'm dealing with some stored procedures that use aliases (I did not write them and I'm new to this project), and they're kind of painful. I realize that the reason I don't like aliases is because of how they're defined. You know how it's generally good practice to declare variables at the top of your scope? (And usually at the beginning of a line?) Aliases in SQL don't follow this convention, which makes me grind my teeth. Thus, I have to search the entire code for a single alias to find out where it is (and what's frustrating is, I have to read through the logic before I find the alias declaration). If it weren't for that, I honestly might like the system better.
If I ever write a stored procedure that someone else will have to deal with, I'm putting my alias definitions in a comment block at the beginning of the file, as a reference. I honestly can't understand how you guys don't go crazy without it.