I doubt I can remember everything but...
Atari 2600: Dragonfire, I think it was called? Amazing arcade game with two screens, one a side scroller where you dodge fire, and another a top down perspective inside the castle, running around and collecting the dragon's treasure. My friends and I used to compete for high scores in various Atari games, and this game was always the highlight of our many competitions.
Intellivision: Utopia, all the way. Far ahead of its time, and one of my favorite games of all time.
NES: Battletoads. I was forged in the Turbo Tunnel.
SNES: I have plenty of fond memories staying up all night with friends playing Mario Kart until our eyes bled.
Sega Genesis: Castlevania: Bloodlines. Sue me, I always thought this game was fucking rad.
N64: I was just having this conversation with my roommate, but I doubt there is any memory more profound, more nostalgic, than running around outside the castle for the first time ever in Super Mario 64. That game was a paradigm shift, and you could feel that the moment you had that controller in your hand.
PS1: I loved Final Fantasy back in the day, and there was no entry I loved more than Final Fantasy IX. Vivi's journey spoke to me on a personal level, in a way no game ever had before. I never imagined I could be so emotionally invested in a game like this until FFIX came along. Looking forward to jumping back in with the PC release.
Sega Saturn: Guardian Heroes is the coolest shit ever.
Sega Dreamcast: No wait, Jet Set Radio might actually be the coolest shit ever.
PS2: I can feel it. I can feel the cosmos. Katamari Damacy changed my life.
Gamecube: I was obsessed with Metroid Prime for months.
Xbox: Phantom Dust is my favorite game of all time. I have no gaming memory more fond than that of staying up all night with my friends, playing Phantom Dust, on the final night of the OG Xbox Live servers. We went hard, until the servers went offline. Truly, the end of my favorite gaming era. A piece of me died there.
Xbox 360: Gears of War multiplayer almost made up for no more Phantom Dust in my life. Almost.
PS3: Demon's Souls. So the world may be mended.
Wii: Super Mario Galaxy was a triumph, probably in my Top 3 Mario games.
PC: Hard to say, but the game I have spent the single most hours of my life playing is probably a toss up between StarCraft, Duke Nukem 3D and Quake 3: Arena.