The Xbox has a very large number of good games out there. I'll list a few from the genre I'm into and link you to the Microsoft counterpart.
RPG
Lost Odyssey
Info
This RPG is one to remember for ages. If you're into any final fantasy game, you will love this. Great character -optional- character depth and game play is fun.
Infinite Undiscovery
Info
As fair warning this RPG has a timid main character.
Action
Armored Core 4
Info
This game is a high paced mech combat game. Great fun building and tweaking these things.
Resident Evil 5
Info
This game is a clunky third person action game. It's in essence a shooter, but the story is well developed and how fighting is really not very friendly, I'd throw it here.-You can turn off mature content-
Strategy
Last Remnant
Info
A great strategy game. You control groups of characters and give them indirect orders as to portray a commander in battle. Be sure to download this to your hard drive as it will reduce issues with any lag while fighting.
FPS
Bioshock
Info
If your into shooters and mystery.. this is a must have. It'll run cheap at the store and will deliver.
Halo 3
Info
The game that made the Xbox. Halo is a base of many FPS games out there.