Personally I think that the Xbox is a under appreciated console. Their are many great Xbox games out there; the Halo series is by far the most famous but many other great exclusive and non-exclusive games exist, including the Far Cry series, the Fable series, the Doom series, Fallout 3, the Driver series, Black, the Mass Effect series, The Chronicles of Riddick, the Dead Rising series, and the Call of Duty series. And games are not all it has going for it, personally I just love how a Xbox (original or 360) controller fits into my hand, I think it is the pinnacle of comfort and control and is really only given a run for its money by the keyboard and mouse.
While I have not played many Xbox 360 games, I did have a Xbox that I played a lot of and I have played many ports of Xbox 360 games on the PC. Of the original Xbox, I really loved The Chronicles of Riddick for its great stealth action, Black, Far Cry, and Urban Chaos for their great and somewhat unique FPS action, Beyond Good and Evil for its great action platforming and story, Armed and Dangerous for it hilarious humour, Syberia for its amazing adventure and story, and Ninja Gaiden for its extremely challenging combat. All of these games were great fun to play and left a lasting impression on me and my opinion of gaming. Of what I have seen of the Xbox 360 it has a far more diverse library of games; From great RPGs like Mass Effect and Dragon Age: Origins through the numerous and ever popular FPSes and sports games to action adventures like Bionic Commando to even a few puzzle/non-action games.