Nice thread.

NES - Super Mario Bros. 3

Genesis - Sonic the Hedgehog

Super NES - The Legend of Zelda: A Link to the Past

Nintendo 64 - The Legend of Zelda: Ocarina of Time (a tie with Super Mario 64)

PS1 - Metal Gear Solid

PS2 - Dark Cloud (What can I say? I really enjoyed that game, it was awesome)

GameCube - Baten Kaitos: Eternal Wings and the Lost Ocean (Absolutely beautiful game. One of my favorite games of all time.)

Gameboy- Pokemon Red (You know it's true.)

N64 - Ocarina

Gamecube - Melee (the best one, smash balls are fun but got old fast.)

PS1 - Metal Gear Solid (It's godly, didn't know that kind of crap was possible on PS1)

PS2 - Shadow of the Colossus, Kingdom Hearts 1 and 2

PS3 - sorry, but MGS4. I have so much faith in how good this game is, that I think there is a very good chance that nothing will be better in PS3's life. Hopefully I'm wrong!

PC - Warcraft 3 Reign of Chaos/Frozen Throne - Don't get me started.  I f*ckin love this game. The story was great and led to WOW.  The online is fantastic, I still play.  Then again it is a Blizz game, people still play Diablo 2.

Nintendo:

NES: Super Mario Bros. 3;
SNES: Super Mario RPG: Legend of the Seven Stars;
N64: Star Fox 64;
GameCube: Metroid Prime;
Wii: Super Mario Galaxy;
Game Boy: Super Mario Land;
Game Boy Color: The Legend of Zelda: Oracle of Ages;
Game Boy Advance: The Legend of Zelda: Minish Cap;
Nintendo DS: Contra IV.

Sony:

PS1: Xenogears;
PS2: Gradius V;
PS3: Metal Gear Solid 4.

Sega:

GEN: Sonic II;
Saturn: Radiant Silvergun;
DC: Sonic Adventure.

Microsoft:

Xbox: Halo: Combat Evolved;
Xbox 360: Mass Effect.

To be updated!!! =P

I'm also throwing in some hand held games if you don't mind...

PC- There are so many to choose from, but I'll mention Grim Fandango because I don't think any body else will.
SNES - Loz: AlttP
N64 - Legend of Zelda: Majora's Mask - Everything about this game I loved. From atmosphere to the great N64 Zelda game play
GC - SSB Melee - A competitive master piece. Once you start playing this game on a competitive level you can really start to appreciate how deep the game really is.
DS - Pokemon Diamond/Pearl - Another competitive marvel. A complex meta game that takes the deepest of thought and concentration to master.
GBC - Zelda: Oracle of Seasons/Ages
PS1 - MGS
PS2 - Another hard one to choose from. But I'll mention Resident Evil 4 because knife runs in that game are so fun. Silent Hill 2 is also another noteworthy game ^__^

Shadow of the Colossus.

Really. That's all that needs to be said.

PC- Portal: Short, but some of the best 4 hours you can spend in a game
PS1 - Suikoden II: I've spoke of this game too much. It's the best game ever made, that's all you need to know
PS2 - Shadow of the Colossus: Epic. Magnificent. Impressive. Outstanding

My picks would be...

PC: Planescape Torment, bar-none.

NES: Super Mario Bros. 3, for obvious reasons, but Kirby's Adventure comes a close second.
SNES: Zelda ALttP. A game that proved that you didn't need to be an RPG or adventure game to tell a coherent story.
N64: Ocarina of Time, obviously.
Gamecube: Killer7. A surreal mindfuck of a game.
Wii: Super Mario Galaxy. I still stand by my opinion that it's the most beautiful game this generation, and it's a pure joy to play to boot.

PS1: Oddworld: Abe's Exoddus. Just plain awesome in every way.
PS2: Lots to choose from, but ICO would be my pick.
PS3: Don't know, as I don't own one.

Xbox: Tough choice, but I'd say Call of Cthulhu: Dark Corners of the Earth.
Xbox 360: Tie between Bioshock and Rez HD.

Also, ugh @ the MGS worship. What do people see in these games, anyway?

PS2: Xenosaga I-III = unmatched emotionally storytelling
Snes: Super Mario World = a masterpiece in many ways,but you can actually feel how Miyamoto said to a programmer for example: 'No this Goomba must beginn to walk one nanosecond later' = absolutely masterpiece of precision which i think is a special kind of art :)

NES: The Legend of Zelda
SNES: The Legend of Zelda - A Link to the Past
N64: The Legend of Zelda - Ocarina of Time
GCN: Metroid Prime (C-c-c-c-combo Breaker)
Wii: Resident Evil 4: Wii Edition
Gameboy Color: The Legend of Zelda - Oracle of Ages
Gameboy Advance: The Legend of Zelda - Minish Cap
Nintendo DS: Phoenix Wright - Trials and Tribulations
Xbox: Halo - Combat Evolved
Xbox 360: Bioshock
PC: Half-Life 2

Pokemon Red / Blue