I started gaming with our SNES, and my brother's old NES for a period. We had about 10 or so games for each, games like Super Mario Bros 1 and 2, Super C, Boulder Dash, Joust, and several others. For SNES we had Super Mario Kart, Super Mario World, Super Off Road, Faceball 2000, Gradius III, Super Punch Out, Yoshi's Island, Uniracers, and a few others.
My brother got a Genesis one Xmas because some neighborhood kids had one. We only ever got a few of the Sonic games, and the SEGA version of Aladdin which was pretty good, maybe a couple of other games but sadly couldn't really afford the time and money to play much with it while the SNES had us occupied. At some point, my father randomly brought home an Atari 2600 someone had given to him, it amused us for all of 15 mins.
Then we got super hyped up for the N64 release. We got it Xmas of 96' and were glued to the thing 24/7. I became something of Ninty fanboy, and looked down on anyone who'd consider buying a PS1 because those graphics. Evidently I thought muddy was better than jaggy. N64 was multiplayer fun like never before. We had Super Mario 64, Mario Kart 64, Banjo, Goldeneye, Mario Party, Duke Nukem 64, Bomberman 64, Cruis'n USA, San Fransisco Rush, Top Gear Rally, and quite a few others. I did get a GameBoy color in 98', but sadly it couldn't hold my attention.
Broke my loyalty to Nintendo by 99', first getting PS1 and then in September of that year, a SEGA Dreamcast. For PS1 I had Twisted Metal 2, Driver, Tony Hawk, Runabout 2, Duke Nukem Time to Kill, and several others. For Dreamcast I had Sonic Adventure, Crazy Taxi, Blue Stinger, Speed Devils, Shenmue, and maybe a few others. But once Phantasy Star Online became a big thing, my brother pretty much took over the Dreamcast and played only that game 24/7. It was the gateway drug that pulled him away from console games and into more hardcore MMORPG games for PC. I never got into that kinda stuff. Instead I contemplated buying either another Dreamcast, or just springing for a PS2. I went with the PS2. Theme Park Rollercoaster was my first game, but also had Midnight Club, and later got into the GTA games, the THPS games and all of that over-rated kinda drivel for a bit. I did eventually pick up another Dreamcast, but stupidly didn't take advantage of the game liquidations because *gasp*, you can just burn the games!
I was lucky enough to have very generous parents who also gifted us both an original Xbox and GameCube, but like a ungrateful turd, I basically paid almost no attention to either once I was hooked on the PS2. I did eventually get a GBA SP, but again, it wasn't able to hold my attention for longer than 15 minutes. That was around the time I first became interested in Saturn, and actually went and bought a second-hand Saturn back in the dark ages of eBay, when there were no pics of what you were buying, and no guarantees. I was never able to find much, or anything cheap for the system even back then, so I kind of gave up on it for the time being.
So that pretty much wraps up the memories from childhood/adolescence for me.
EDIT: Well, I guess there was some PC games that I played. Roller Coaster Tycoon, and The Sims is basically all that needs to be said. We had a Compaq Presario back in the day, and man it was terrible for running games.