For me:
SNES: I was like 4 when this came out, and I don't remember gaming beforehand but if I did it would have obviously been Super Mario World
Genesis: First console I remember wanting, got it in 1992. Needed Sonic the Hedgehog 2. First game I ever beat.
Saturn: Years after its release but it was Panzer Dragoon and Guardian Heroes that made me want one. I also wanted Sonic 4 which never came >
PlayStation: I was a huge platformer fan so I got it for Crash Bandicoot, but Final Fantasy VII is also what made me get it
Nintendo 64: Got it during launch year for Super Mario 64
PS2: Got it during Christmas 2002 with Vice City and Shinobi. Wanted those games so bad. Also got WWE Just Bring It which was pretty good
Xbox: Halo. The first $350 game.
GameCube: Don't remember specifically but probably Melee
Xbox 360: Halo 3 and Gears of War, but I got it launch day so for then it was Call of Duty 2 and Kameo
Wii: Only console I have waited in line for launch. Got it for Wii Sports and Twilight Princess
PS3: I don't know why I bought this at first cause I didn't want any games for it til around 2008 (got it early 2007), but MGS4 was my big game
Handhelds: DS for Nintendogs, GBA for Pokemon, GB for Link's Awakening