Just a quick list of some of the good and bad things about each off the top of my head.

xbox is the most powerfull of the 3.

gamecube is the least powerfull.

ps2 falls somewhere inbetween in terms of computing power.

xbox and ps2 can also play dvds.

ps2 has the more games than the others.

ps2 is backwards compatible with ps1 games.

gamecube is the cheapest.

gamecube is the smallest and most portable.

xbox has a built in hard drive (no need for memory cards unless you want to play your game on a friends box. Plus you can rip cd's onto the hardrive and use it as a jukebox.)

gamecube has the best controller for most people.

xbox is the most easily modded.

My personal recomendation is that you buy the console that has the games you like the best. All 3 of the have some games that are oriented towards kids and some that are more for adults (gamecube probably has more for kids than the others). They are all fairly good at what they do and are a lot of fun. If one of them has a game or 2 that you are dying to have, then you should get that one. My personal preference is for the ps2 just because i like more of the games it has than any of the others. But, like Russmeister said, I would recomend playing them before you buy.

So to sum it all up:
if you're looking for features, go with the xbox.
if you're on a limited budget or want something portable, go for the gamecube.
if you want the widest selection of games, then its the ps2.

And now I've gotta go take my last final for the semester. Wish me luck