These could be games that left a lasting impression on you, games that had stellar gameplay mechanics, characters that captivated you, games that you played tons of hours on, etc.
Megadrive - Gain Ground, Sonic 1&2, World of Illusion
PS1 - Toy Story 2, Rayman 1&2, FFVII, Rugrats, Tony Hawks 1-4
Gameboy/Color - Pokemon Blue, Tetris, Rayman, Tweety - Around the World in 80 days
PS2 - Kingdom Hearts 1&2, GTA 3,VC&SA, Need for Speed Undeground 1&2, Tony Hawks Underground & American Wasteland, Jak 1-3, Sims Bustin Out
GBA - Pokemom Fire Red
Xbox - Halo 2, Need for Speed Most Wanted, KOTOR 1&2, Fable, Prince of Persia Warrior Within, *Fahrenheit
Gamecube - Wind Waker, Sonic Adventure 2, Mario Sunshine, Luigis Mansion (I didnt have a gamecube as a kid but was enamroed by these at a friends house)
DS - Mario 64 DS, Pokemon Diamond. New Super Mario Bros, Worms Open Warfare, Sims 2
360 - Rockband 1-3, Guitar Hero World Tour & 5, Halo 3, Battlefield Bad Company, Mass Effect, Banjo Kazooie Nuts and Bolts, Far Cry 2, Assassins Creed 1-3, GTA IV
PC - Rome Total War, Empires Dawn of the Modern World, Black & White 2, The Movies, Sims 1-2 (I liked 3 too but I was older than a kid then), Spore
Lines are a bit blurred at a certain point for 360 and PSP to when I wouldve no longer been a kid and more of a late teenager or whatever.
As an adult I've looped back around to the DS. Never really appreciated it as a young teenager but it has an amazing library. Ace Attorney and Devil Survivor have racked up a lot of play time from me in the past few years.