I'd say these are the 5 most impactful games in terms of defining my tastes in gaming:
Super Mario Bros. (NES) - This was the game that really got me hooked on platformers. It was also the game that introduced me to the NES. Prior to that, I was playing Atari 5200 and 2600, which I enjoyed, but gaming was more of a casual hobby for me then. That all changed with the NES; gaming became more of an obsession. Super Mario Bros. was addictive, challenging (at the time), had catchy and memorable music, and was set in a fun, but trippy world. Platformers are still one of my favorite genres today.
Mega Man 2 (NES) - This was the game that introduced me to the Mega Man franchise, which would go on to become my favorite franchise. I remember some my friends used to complain that there were too many Mega Man games and that they were too similar, but I didn't care. I couldn't get enough Mega Man! It combined platforming with a variety of unique weapons, lots of fun bosses, and fantastic music. Mega Man 2 was really the game that got me interested in video game music. To this day, it is still my all-time favorite video game soundtrack. The fact that it was composed on the NES, using only the 4 standard PSG channels (2 rectangles, 1 triangle, and 1 white noise) is pretty impressive. Modern consoles allow for much more variety in instrument sets, since the music is generally just streamed, rather than executed as program code. However, for me, nothing beats the Mega Man 2 soundtrack, despite the technical limitations under which it was composed.
The Legend of Zelda (NES) - This was my first action/adventure game, which remains one of my favorite genres. It was the first game I played that involved any significant exploration.
F-Zero (SNES) - I had enjoyed playing racing games since the Atari days, but F-Zero was the first one that really wowed me. I just loved the speed, the soundtrack, and the futuristic setting. Racing is still one of my favorite genres and F-Zero is my favorite racing franchise. I'm not really a big fan of simulation racers, but I love arcade racers and I think F-Zero is a big reason for that.
R-Type (ARC) - This was the game that got me hooked on SHMUPs. I used to play this game at a local pizza place. Prior to playing this, the only SHMUPs I had played were the single screen variety (Galaxian, Space Invaders, etc.). Those were fun, but R-Type just blew me away. At this particular pizza place I went to, you could bring in your report card and they would give you tokens for each A or B you got. I would spend a big chunk of those on R-Type. SHMUPs are still one of my favorite genres, even though they seem to be in decline.