Adventure: Sherlock Holmes and Jack the Ripper is an awesome Adventure game, but the 1st place goes to... Battle of Olympus.
Other - Legend of Zelda - series.
Arcade: King of Dragons and Final Fight will have to be my number ones.
Other - Breakout.
Fighting: Street Fighter II. Awesome music, awesome game.
Other - Street Fighter IV, Dead or Alive 4, Final Fight, Streets of Rage.
Platform: Mega Man-series. The game is as awesome as the music!
Other - Super Mario 64
Puzzle: Sorry Box, but I'll have to say Tetris.
Other - Draculas Riddle I count as a puzzle-game, but other games are Solomons Key and Lolo. Qwirk/Kwirk for the GB is an awesome game as well.
RPG: FINAL FANTASY SERIES! FF FF FF FF FF FF FF FF!
Other: Chrono Trigger, Secret of Mana, Secret of Evermore, Breath of Fire, Suikoden, Vandal Hearts, Dragon Warrior, Dragon Quest, Eternal Sonata (Best music ever)
Racing: PGR 4.
Other: Daytona USA, Outrun, RC PRO AM, Carmageddon
Shooter: Counter-Strike Source and Quake Live is my games of choice.
Other: Call of Duty IV, BF:BC2 and Quake 2.
Sports: FIFA 10, Tiger Woods PGA 10, Top Spin 3, Super Tennis, Super Soccer, Tecmo World Cup Soccer, World Cup (NES)
Other: NHL, Table Tennis, NBAgames
Strategy: Sengoku Rance! (Hentai-game but if you look past the hentai, which I do, it's a superfun game.) Also, I like Civilization!
Other: Utawarerumono. (also a hentaigame but look past the hentai and it's awesome.)
Multiplayer: Hmm... Quake Live!
Sandbox: San Andreas! / Prototype!