When I was born - 1983 - We had the Atari, on which I played Centipede, Asteroid (I think), Pacman, and such. I barely remember this. But my parents told me.
1986 - Dad bought himself the Nintendo Entertainment System, which immediately became mine. This is where my gaming addiction took off and never subsided. One of my fondest, earliest memories with my father was the two of us playing through the original Legend of Zelda together. I was so young, he did most of the thinking and helped me a great deal. But we ended up beating it. I think we even did the second master quest, but I'm not sure. Also was heavily into Super Mario, Donkey Kong, Castlevania, and of course, Super Mario Bros 3. Also I loved the game Kung Fu, I still remember the way the first boss would laugh when he'd kill you, as does my mom, who heard the sound very often. I think last time I counted I had around 40 games for the NES. The NES lasted me until around 93/94, when I got....
1993 -Super Nintendo - The best system of all time, imo. This had some of the most immensely amazing games I had ever played, including most of my all time favorites, such as Final Fantasy VI, Tetris Attack, Chrono Trigger, and so on.
Final Fantasy VI was the first game that I was really addicted to. It was the first game which I would sit and play for 13 hours straight and such. It was immediately the best game I had ever played, still is, to this day, my favorite and most cherished game of all time.
The game Killer Instinct holds a place in my heart because I remember to this day how excited I was when that was going to come out. The night before it came out I literally couldn't sleep, I was too excited. That day, my parents and I were planning on going to San Diego for our family vacation. I talked my dad into letting me bring the Super Nintendo with us so I could play my new game. We were planning on leaving at around noon, buth there was a problem. The shipment for Killer Instinct hadn't arrived at the local Toys R Us. My dad said we could wait till 1:00 or so, but then we had to go whether I had the game or not. I called the store every 10 minutes asking if they got the shipment to no avail until about 12:45. Then my dad said we had to leave, so I called once more, and to my immense joy, they had just received the shipment. The lady who answered all my calls said she was so happy that the shipment came because it meant that I would stop calling. I laughed and thanked her. We then picked it up and went straight to San Diego for our vacation, which consisted of me playing Killer Instinct constantly. My Dad was upset and regretted letting me take the system because I didn't want to do anything else, but whatever, it has since become an important memory.
This system lasted well through 96 (or whenever the N64 came out), and still go back and play it sometimes. Just about 6 months ago me and one of my friends really got back into Tetris Attack, which we played for about 2 months almost every day. That game is amazing.
In 1995 my parents bought me a Sega Genesis for some reason. I never asked for one or even showed interest, but they thought it would help the wait for the N64. It was a nice system, I remember enjoying the Sonic Games, but I got it to late and was still very much into the Super Nintendo, so I didn't experience some of the Sega classics.
1996 - N64 - The first system I bought for myself, with money I had saved up since the announcement of the system. I reserved it months before it came out at Toys R Us. About 3 weeks before the system was set for release, the local Blockbuster got one in and set it up for public play. Every day after school until the system came out, my friend and I would walk over and play it for several hours. Usually we had to just replay the same segments of Super Mario 64 over and over but I didn't care. Sometimes the (understandably) annoyed employees would kick us out after an hour or so, but usually they didn't care.
Then on the Friday before the system was released I heard rumors at school that Toys R Us got shipments early and were selling them to customers who preordered. I nearly shit my pants, and when I got home I called, and sure enough they had released them early (which got them in trouble with Nintendo, but who cares?). I went and got my system and proceeded to play it all weekend. I don't know if you guys have seen the South Park episode, where the kids get a new system and start planning their days around it and calculate how much time they have to play before they have to do something that they don't want to do, but I quite literally did that exact thing - (Okay, I just got out of school, I have a good 8 hours to play before I have to go to bed, maybe 9 hours if I can talk my mom into letting me stay up late).
The excitement and anticipation for this system was unparalleled. Of course, it had some of the greatest games of all time - Goldeneye, Ocarina of Time, Killer Instinct Gold, and so on. I adore this system.
In 97 I got a Playstation, along with FFVII and NHL98 for Christmas. Both wonderful games. I later got other classics such as Metal Gear Solid, Gran Turismo 1 and 2, and Resident Evil, another massively great system, which totally deserved its popularity.
In 99/2000 I got heavily into PC games because my dad bought us a new computer, specifically Half Life and Counter strike. This caused my console interest to decrease during the next generation, I got the PS2 and Gamecube, but didn't play them nearly as much as I had prior systems. The PC pretty much dominated from 2000 - 2006.
In 2006 I got the 360, which is currently gone for repairs. And a few months ago I got the Wii. I will get a PS3 someday, whenever there are more games that I wish to play. As it is, gaming memories are some of my fondest, and it's one of the few interests that have stayed strong throughout my life.
I remember back when I was a kid and playing N64, the adults would tell me that it's just a phase. That when I got older I would loose interest in gaming and concentrate on girls and other things.
Needless to say, it never came true, I still love games and believe I always will.
Also, I forgot to mention handhelds. I was never really into handhelds until I got the DS Lite. I had some though - The first being Gamegear, which was a nice but flawed system. I'm a bit saddened to say I never owned an original Gameboy. I know I missed out on some amazing games, which sucks. I did get the great Gameboy Color, though. Then the Gameboy Advance SP, and finally, the DS and last year the DS Lite.
I also bought the Dreamcast at some point, but only played a few games on it, though they were great - Grandia II and Marvel vs Capcom II.
Currently, I rotate between the PC (World of Warcraft, mostly) and the 360 and the Wii.