If offered the choice, I would definitely take the PS2 or the Gamecube over the Xbox in a hot second, but I do still have a fair amount of affection for gaming's girthiest console. In fact, these days it's one of my favorites to collect for since, for the most part, its games are dirt cheap.
Like several of you have already said, I was introduced to the Xbox in college via friends that were really, really into Halo. I liked Halo well enough, but I lived in a dorm with a bunch of dudes who were incredibly competitive, so eventually I just kind of moved on from it after I got tired of all the alpha-male posturing. About a year or so later, some wealthy relatives bought my brother and I each an Xbox for Christmas, and I ended up grabbing a copy of Halo since it was really the only game on the system I knew anything about. Eventually my brother (who amusingly enough now works for Bungie) and all of his friends got super into Halo, prompting me to give up on it once again. Towards the end of college, when I started to first dabble in game collecting, I did a little bit of poking around the edges of the Xbox library and found there were some really weird games out there.
I played the hell out of Otogi, which to this day I still consider an action game masterpiece (I like the second one too, but I think I prefer the simplicity of the first game). Breakdown was a fascinating first-person brawler/shooter/horror hybrid that I'm still not quite sure I understand from a logistical standpoint. Speaking of unique first-person games, The Chronicles of Riddick: Escape from Butcher Bay was a really great, really unique action/stealth/puzzle game that introduced me to Vin Diesel's Riddick character and his sci-fi universe. And of course, there were some brilliant Sega gems like House of the Dead III and Panzer Dragoon Orta as well (I know Otogi was published by Sega, but I really consider it a From game more than anything else).
After college, being a big Bioware fan, I gobbled up the two Knights of the Old Republic games, although I skipped over Jade Empire for some reason. I really enjoyed the Jedi Knight ports for the system too. Gun was a weird little spaghetti western homage (that I think got a 360 release too) that I thoroughly enjoyed but never hear anyone talking about. And I absolutely fell in love with TimeSplitters: Future Perfect, which to this day I consider one of my favorite games of all time. I even did
a Let's Play of it. Speaking of Let's Plays, I did one of
Stubbs the Zombie too, another criminally underrated Xbox action game with significant strategic elements, made by some of the former Halo crew. It got ported to modern consoles recently, which kind of blew my mind, since it had seemed totally lost to the ages until then.
A friend of mine was a big Sudeki apologist, so he grabbed me a copy of that when it was, like, $5, and I will say I did genuinely enjoy it. It's not a masterpiece, but it is a fun, little RPG. And I played through a handful of multiplatform games like Beyond Good and Evil, Lord of the Rings: The Return of the King, and TimeSplitters 2 as well, all of which were lots of fun.
I think I said it in the 360 Console of the Month thread, but the original Xbox was one of the last consoles that I really felt had a personality to it, seeming to have positioned itself as the torchbearer of Sega's console 'tude, with a little bit of America's "Bigger is Better" mixed in for good measure. Not that we should necessarily be ascribing personality or values to pieces of commercially released hardware, but it is what it is. As a piece of hardware, I'm not the biggest fan because, despite its malleability, it is fairly large and clunky, and as markies said, prone to failure. But it definitely has an interesting library, and despite the general view that everything is a PC port, it had a significant number of very solid exclusives that haven't ever been ported anywhere else. If anyone has any recommendations for (preferably less expensive) exclusives worth picking up, I'm all ears.