I know that Keith Richards has been known to use open tunings on some songs, but what is an open tuning exactly and when/why would you use it?
As Ian said, an open tuning is one where the open strings form a chord.
Having a chord available on the open strings is very useful for slide playing (unless you're REM or Steve Cropper (who often uses a Zippo lighter as a slide, so he really doesn't need a full open chord)) since it gives you chord intervals among the strings at the same fret. So for slide blues you can tune the open strings to the root chord (I); then your subdominant (IV) is at the fifth fret and your dominant (V) is at the seventh fret (with chromatic mediants at the third, fourth, eighth, ninth, and tenth frets).
Open tunings are also useful for "folk" guitar styles as they can give you many nice drone strings to beef-up a simple fretted melody. And for rock styles they can offer interesting possibilities for partial chords (I think this was the big draw for Keith Richards).
My favorite tuning is D-G-D-G-B-D, which I feel compelled to mention as it was omitted from Ian's list (otherwise excellent). It's Open G. Why use it? It's better than Open D for almost everything. You can slide into the root from below (unlike Open D). You get a minor third at the top (nice for sliding into 5-b7 part of the chord) (also nice for fast melodic runs by puting shorter intervals on the higher strings (where your fingers are curled) and longer intervals on the lowers strings (where your fingers are already stretched out). And you can still do one-finger power-chords, but you have to do them on the "A" string (standard). And it's easier to return to standard (than with D) because you've only altered 3 strings (D)E-(G)A-D-G-B-(D)E (not 4 (D)E-A-D-(F#)G-(A)B-(D)E ).
Open G also gives you "more" Octaves. If you play a little melody across a few strings, it is much easier to find the octave transposition. The Open chord is very "5"-heavy giving it an unstable, edgy effect.
For a while I dropped the whole thing down a whole step producing C-F-C-F-A-C. But then, everything's too low to sing properly. But I often still write with a C-clef which was quite natural for this tuning.
An "open tuning" is one where the open strings, when strummed, are tuned to form a chord. So an "open D tuning" is going to something along the lines of (low to high) D A D F# A D -- that's a D major chord.
There are certainly more common open tunings than others but the range of open tunings available, even with a standard set of guitar strings on a standard scale guitar, is pretty large when you take in to account more than just major and minor chord shapes.
From this Wikipedia article, some more common open tunings would be:
There's quite a few more, including some suspended open tunings and 1st-and-5ths only tunings that make for some cool acoustic tunes. Check that link.
I was at a clinic with Don Ross last year and I believe he said he was up to the 40 or so open tunings range now. Granted he uses a long scale guitar to help extend the range of notes he can open tune to. He tries to plan his sets around tunings so he doesn't have to change many strings from song to song.