Why exactly would you want an alternate tuning? Does it provide a different tone or just make it easier to form the chords you want to form? Does it actually make a different sound than standard tuning - in other words, can you always duplicate something played with some other tuning on a standard tuned guitar?
Am I even going to understand the answers to this question?
I don’t have a lot of time, but here’s a little bit:
you can transfer stuff you play from one tuning to another depending on how that new tuning requires you to stretch your fingers!
different tunings are typically (but NOT always, by any means) more about tuning the strings so they make a chord when you strum them open. Standard tuning is a compromise; it makes playing in most keys pretty straightforward. Most alt tunings basically say “screw that - let’s make it super easy to play in one key; all the other keys be damned” 
so you end up being able to play chords with one finger - just fret with one finger up the neck and you’re in business. This can be VERY useful for slide, since you only have one surface (the face of the slide as you lay it against the strings) with which to make chords.
Having the strings form a chord in the open position also sounds different vs. standard tuning - the harmonic overtones that layer over the main string tones vibrate more sympathetically because the strings are in a chorded relationship to one another already, if that makes sense. So banging on a major chord in a open tuning can sound much bigger and richer. Keith Richards exploits this better than anyone - when you hear the signature openings to Start Me Up and Brown Sugar and wonder “why are those chords so instantly recognizable?” it’s partly because he is playing in Open G and partly because he’s Keith Fucking Richards!!
Other alternate tunings that DON’T form chords when you play them with one string simply provide different fingering options and harmonic overtones. Jimmy Page’s use of DADGAD is like this…think Kashmir.
All I got for now…
I think Wordman’s post covers almost everything, but…
One great thing about alot of alternate tunings is that you can easily play the exact same note on two strings of the guitar. It really doesn’t sound like a big deal, but it can make a guitar part sound much more full than it otherwise would. A great example of this would be the song Venus in Furs by Velvet Underground. Every string on Lou Reed’s guitar is tuned to the same strings, so it ends up sounding like 2 or 3 guitars playing the same part.
Anyone who cares about alternate tunings should spend some time listening to Sonic Youth, who almost never play guitars that use standard tuning. They really have a that full resonating sound in every song.
The most common alternate tuning is the “Drop D” tuning… The low E string is tuned down to a D.
With that, the three low strings form a “power” chord (Root/Fifth/Octave) and can be barred up and down the fretboard.
There’s another kind of alternate tuning: “step-down” tunings, in which you drop all six strings the same number of half-steps. It’s sort of the inverse of using a capo. The advantage is that is, as with the capo, it lets you use the familiar finger positions to fret different chords, changing how they sound and in some instances making them easier to play.
Probably the best known instance of this is “Yesterday,” which Paul McCartney plays with his guitar tuned down a whole step, allowing him to play a progression in F using G chord shapes. Kurt Cobain wrote and played many Nirvana songs tuned a half-step down.
edit: fixed spacing
A lot of Joni Mitchell’s songs are in ‘open tuning’. DADFAD, I think is one, if memory isn’t faulty. You get some very, very pretty and complex chords without having to do difficult fingering. The sound is a little different, too, I think. Simple country or rock songs seem to use a lot of straight D, G, A, E, F, C chords. With an open tuning, you play something like A minor suspended fifth. (I totally made that up, sorry.) So the sound of the guitar played in an open tuning isn’t as ‘basic’ to my ears.
Note: I haven’t really played guitar in about 14 years, so I lack much credibility and salt should be applied to post.
If I remember correctly, quite a few of the harder rock groups (back in the day) used to actually tune down as much as a minor third, for the exact reasons already mentioned. Also, when writing songs, sometimes you find yourself in the frustrating position of coming up with a fantastic vocal melody, with perfect sounding chords, that you just can’t friggin’ sing high enough to make work in standard tuning. Step-down tunings can be a big help when/if that comes up.
In recent years, Drop-D tuning has been in vogue with many punk/emo groups, largely because it gives you a nice fat power chord simply by laying one finger across the bottom two strings. Whether this is “stylistic” or “just plain lazy” is in the eye of the beholder, I suppose 
Anyway, let me take this opportunity to geek out for a sec…
One of the things I love about Drop-D is that it gives you a terrific open D chord with a big bottom end. However, if you play a song that also involves barre chords or something else further up the neck, you sometimes have to involve fairly frustrating hand positions. However, if you have a clip-style capo (not the wrap around kind… yeah, I know a lot of people like those better, but they don’t work for this trick), you can have the best of both worlds. Put the capo on “backwards” (i.e. jaws of the clip facing the lowest string rather than the highest), but only cover the top five strings. Presto - strings 2 through 6 are now up a whole step, but string 1 remains the same. This gives you the same benefits as the Drop-D when playing open chords, but leaves barre chords and other things higher on the neck unchanged.
Man, I wish I’d known this back when I used to while away the hours playing along with Beatles records, wondering “What guitarist in his right mind writes a song in freakin’ F?”
A LITTLE different ? Don Juan’s Reckless Daughter or Hejira are tour de force in this regard though the direction of composition highlights the novelty.
A quick reply - one changes the tuning to make particular fretting-hand patterns easier.
Classical guitarists often tune the 3rd string to an F# when playing lute or vihuela music, as it seems to reflect what the tuning was on the original instruments. Bach and Weiss are played on guitar in either tuning by different guitarists, and the only consensus is that the pieces are challenging no matter how you tune the instrument.
Stanley Jordan uses a tuning that is in straight 4ths all the way across the board. There is also a tuning that uses straight major 3rds all the way across the board. The advantage of these tuning systems is that everything you play is in the same fingershape regardless of what strings you’re playing. The disadvantage is that it makes it quite difficult to play open ‘folk’ chords.
There’s something about alternate tunings that particularly disturbs pianists - the idea that the shape that usually produces C Major can be altered so that it produces c minor drives them nuts! It’s hard enough for people used to the keyboard to accept that the same notes require different fingershapes depending on which strings you play them on, and the idea that you can play a higher note on a lower string is a bit of a mind-bender, but the idea that you can change the tuning of the instrument makes some pianists feel like they’re being asked to play an octave with a black key and a white key.
Another good popular (well, since that VW ad) example is Nick Drake’s Pink Moon. It sounds richer and compliments his dusky voice nicely.
It’s perhaps worthwhile to also see guitar tunings in the context of retunings of other string instruments - ‘scordatura’ is the term used in classical music.
I’ve played both (though I haven’t touched piano in a several years) and do have some issues like that. I’ve experimented with alternate guitar tunings, but find it disturbing in a vague way. So perhaps my piano background is stifling me here.
The thing I really can’t get my head around is some instruments like sax, harmonica, others, where the instrument is said to be a B-flat or some other specific key. What, there’s notes that can’t be played at all? No, its the fingering that is more or less difficult for some keys on some instruments, I’m told. I guess I really can’t “get” that idea, unless I played that instrument enough to be as familiar as I am with piano and guitar. Which isn’t going to happen.
Yeah, I’m from a piano background myself and the guitar in general is a strange and foreign object to me. You can play the same note in a billion different places? That’s screwed up and no doubt about it. And then you can go and change your instrument so it plays different notes? Unnatural.
Actually, a sax being “in B-flat” and a harmonica being “in B-flat” mean different things. In the case of the sax (or clarinet, trumpet, etc.), it means that the instrument is a transposing instrument, sounding a major second lower than written. We just recently discussed the historical reasons for this here.
A diatonic harmonica in B-flat does indeed play only the notes of the B-flat scale–and even then there are gaps. The missing notes, if desired, can only be played by using note-bending techniques.
This also makes the strings easier to bend, which I think is the reason Jimi Hendrix used it. As far as I know, almost all of his songs and all of Nirvana’s are in E flat tuning.
Although I’ve read that Machine Gun is in C#, which is so detuned it’s close to rubber. Which helps explain the watery sounding guitar in some parts of the song.
Aha, Thanks. I was conflating the limitations of a sax and harmonica, sorry. Reading that other thread, I see I had forgotten some of the limitations of brass & reed instruments.
I remember a time when I sang a Debussy song - Le Jet d’eau. The original key is C; I needed to sing it in Bb. Transposing Debussy is just not a good idea, and this song proved that point admirably. There was a recurring motif that involved using the thumb to finger C+D and the pinky to play C+D an octave above. This motif would then shift all over, but clever Claude had always made it so that it was easy to play both notes with the thumb followed by both notes with the pinky - always white notes or always black notes, but never a mix. In Bb, that motif became Bb+C, to Bb+C an octave up. My pianist didn’t want to kill me until the show was over, because that’s when he played the original and found out how tremendously much harder my key was…
Later, on an electronic instrument, he played it ‘transposed’ down a major second by using the little button. The sound quality of the instrument was nothing better than passable, but the phrasing suddenly worked.