Shooter(FPS):
Unreal Tournament (low specs), unreal tournament 2k4(med specs), or Unreal tournament 3 (mid-high-ish specs).

Adventure(RPG):
Fallout 1(low specs), fable or Diablo II(mid specs), ummm… Bioshock? (high-ish specs)

Racing:
Need For Speed III: Hot Pursuit (low specs), GRID and DIRT (mid-high-ish)

Real Time Strategy (RTS):
Starcraft (low Specs), Command & Conquer: Generals (mid specs), Command $ Conquer 3 (high specs)

SandBox/Free Ranging:
GTA: 3 and GTA: Vice City (low Spec), Garry’s Mod (Mid Spec), GTA: IV (high spec)

Sorry if some of this sounds patronizing (like saying RTS after Real Time Strategy, It’s not targeted at you in particular, but it’s good info to have here for future reference.

If your wondering if you can run one of the games, head here and use their java applet to check:
http://www.systemrequirementslab.com/referrer/srtest

Also, If you have any free time, I own most of the games I listed above, so you could check them out before you buy them.

~Peace