Long time lurker, first time poster.
I just got spent some affiliate $ and got a new gaming rig with about 20 games. I've worked my way through about 1/2 of them. So far...
FPS
Call of Duty - You can usually log a few multiplayer games
Left 4 Dead 1 & 2 - The levels last around 30 minutes to 1 hr
Borderlands
Crysis
RTS
Starcraft 2 - Accept no substitutes. Games last 20 minutes usually. 1 hr at most.
DOTA Clones
League of Legends - I second Ben's recommendation. Incredibly deep game. 40 customizable heroes. The "easiest" of the dota clones for n00bs. However be prepared for hostile players if you suck. You need good micro. Twitch & Master Yi are badass. I've uninstalled and installed this game like 3x already...
Racing
Blur - Mario Kart Meets Gran Turismo
DON"T Play Civ5. There is no possible way you will be able to control yourself. I've started games at 11PM and stopped around 2PM the next day, eyes bloodshot, more wasted than after a night of drinking. Oh and that's only 1/2 a game.
If you have a particular genre you like let me know...
Best
Yale