Because I'm a super nerd, I'll add a few categories!
Traditional RPG: Dragon Quest 8: Journey of the Cursed King
Strategy RPG: Final Fantasy Tactics: The War of the Lions (PSP)
Action RPG: Illusion of Gaia (SNES)
2D Platformer: Super Mario Bros. 3 (NES)
3D Platformer: Super Mario Galaxy (Wii)
Fighting: Guilty Gear XX (PS2)
Shooter: Metal Gear Solid 3: Snake Eater (PS2)
Party: Dokapon Kingdom (Wii/PS2)
Compilation: Retro Game Challenge (DS)
Racing: I guess F-Zero (SNES) is the only major racing game I've played a lot of... I'm not much into racing...
Sports: Ice Hockey (NES)
Puzzle: Does the collective Ace Attorney series count?
And while I'm at it, let's cover my faves for each console!
NES: Legacy of the Wizard
Super NES: Tough call, actually. FF4, FF6, Chrono Trigger, Breath of Fire 2, Lufia 2 OR Earthbound, realistically...
Game Boy: Dragon Warrior 3
Nintendo 64: Legend of Zelda: Ocarina of Time
GameCube: Legend of Zelda: Wind Waker
Nintendo DS: Shin Megami Tensei: Strange Journey
Wii: Super Smash Bros. Brawl
PS1: Xenogears
PS2: Shin Megami Tensei: Nocturne
PS3: Valkyria Chronicles
PSP: Half-Minute Hero
Downloadable: Mega Man 9
But I'll spare you all my reasons... unless I guess you WANT to know.

And yes, some of these do conflict. I play a lot of games; I have a lot of favorites.