I have vague memories of playing around with Space Invaders on an Atari when I was really, super young. However the games I really got into when I was little include:
Nintendo: Super Mario Brothers 3, Teenaged Mutant Ninja Turtles 3, and Bionic Commando.
Game Boy: I was all about Zelda on the game boy and that's about it.
The PC was really the main platform for my gaming habits back when I was young. Even today, my main platform is the PC above all else- even the PS3 and the Xbox360. (Though I did, and still do, regret selling my 360 after I heard the announcment of Street Fighter 4.

)
The PC games of my youth include:
A whole plathora of crappy independent games based on things like The Flintstones, the Jetsons, etc.
EGA Trek (Please tell me someone other than me remembers this one)
Doom (Of course, the FPS that started the FPS genre)
Commander Keen
The
Tex Murphy series. (Cut short mid-cliff hanger thanks to Microsoft acquiring Access games.)
The
Journeyman Project series.
Myst
Sim City 2000 (Which I maintain is still the best Sim City game to date.)
Sim Copter (I loved using the cheats to get the Apache gun ship)
Sim Tower
Star Trek: The 25th Anniversary
Star Trek: Judgment Rites
X-wing
Tie Fighter
Star Wars: Dark Forces
Star Wars: Rebel Assault 2
DuckTails
Outpost
I think that about covers most of the major games that I've played in my youth- before the grand era of Half-Life, Halo, and Starcraft and Battlefield.
