Fighting: Speed characters like Millia (Guilty Gear), Cammy (SF2), Sonic (Brawl) are the ones I usually gravitate toward. Though sometimes I would take a slower but more powerful character, like Zelda in Melee (no, not Sheik).
Racing: High grip is usually my main requirement. I like to be able to point the vehicle in a direction and just go that way without sliding all over the damn place.
FPS: Heavy offense, usually with speedier characters (Scout or Pyro in TF2, for example). If there's a choice, I'll usually take a shotgun. Something about them is just...satisfying.
MMOs: Ranged or melee DPS, or a combination of the two if possible. Speed is the name of the game.