A lot of stuff covered already. As far as old 360 games:
Crackdown: Sandbox/superhero-ish type game, loads of fun, a real sleeper. Most people bought it for the Halo3 beta codes but ended up actually liking it.
In the beginning you can barely jump up a curb, by the end you can leap over skyscrappers. There is truly a sense of progression as far as skill acquiring goes.
Fallout3: Morrowind style (w/guns) set in a post apocalyptic DC. Awesome.
Batman Arkham Asylum: It's a batman game. That doesn't suck (a miracle!). Brutal (yet somewhat simplistic combat), fun gadgets, cool boss fights (scarecrow
)
All the original animated series cast returns, with a great performance by Luke Skywalker as the Joker.
Dead Space: Survival horror. In space.
Mirrors Edge: Really fun parkour game with an new/weird visual style. I've actually replayed this one a few times, it's always fun to bounce off walls and run up things.
Lost Odyssey: VERY traditional JRPG, made by the godfather of Final Fantasy (teh gooch). Brilliant story line, I think I cried.
Mass Effect: Bioware style sci-fi western rpg. The sequel has swifter/more exciting combat, but as a whole the first entry is more solid, and in the end a heck of a lot more memorable.
Great story and fun "choose your own adventure" dialogue options.
Orange box. Half-Life. Portal. Team Fortress. Nuff said.
Bioshock: Another game with so-so 1st person shooter/magic power combat but a brilliant story telling experience.
I really enjoyed that the game approached things from an almost academic (albeit Randian) angle, tricking the brainless gamer masses into debating philosophy.
The plasmid(magic/abilitiy/skills) customizations are fun and can really change up how you approach each battle.
Dead Rising: Like Dawn of the Dead the video game. Zombie outbreak hits, and you're trapped in a mall. Awesomness ensues.
) wanna play a game sometime? I'll do some practice first though 
