I had the exact same question when I had just started playing electric. Basically, I tried one out for a while just because I was trying to play some U2 songs, but after a while, I didn't really understand its purpose beyond spitting out extra noises.

However, after learning from a much superior guitarist, I've discovered that delay is a very useful effect, actually, every guitarist should have this in their arsenal.

You can use it to achieve something called "slapback" delay, which is when the note you play is repeated at a lower volume, very quickly after the original note. You can hear this in many guitar solos.

It can also be used to create ethereal sounds, by combining a long delay time with many repeats and the effect level at the highest setting and by using volume swells. This makes a really cool effect, actually, I believe there's a track on the new Audioslave album that uses this in the intro.

Another use I can think of off the top of my head is to create cool syncopated rhythms, for example, Roger Waters uses a delay on his bass to get the super-cool riff on "One of these days" off of the album "Meddle".

That's just three uses I can think of, I'm sure there are hundreds more (OK, that's an exaggeration), and I'm sure you other guys here can fill in some blanks too.