So easy to put down 5 favorite games rather than 5 games that define me as a gamer. Not sure I can only find 5 games can really define me as a gamer, but let's try anyway.
Combat - this is an old Atari game where you were a tank or plane (or group of planes) and attacking a similar enemy in a obstacle ridden environment. The AI was crap (as most Atari games AI was) but the option for multiplayer was there. I played this game a lot and is one of the first ones I remember playing (along with classics like Pitfall, Pac-Man, Centipede, Asteroids, etc.), which ultimately set me on the path of becoming a gamer.
Star Wars: X-Wing - This was one of my first computer games and began a love of space combat simulators. The graphics at the time were awesome (though they were unskinned polygons), the combat was fun and the missions ranged from easy to incredibly challenging. From this game I went on to the Wing Commander series, later Star Wars titles, Privateer, and pretty much anything that put me in a spaceship. And I am really hoping Chris Roberts' project Star Citizen makes it, as I have missed this genre greatly.
Star Wars: Dark Forces - essentially the first FPS I played. Most began with Doom, or even Wolfenstein 3D (which I had played a bit at a friends house), but Dark Forces is what made me see the potential of the FPS. It was a bit more advanced than Doom in its options, just as look up and down and crouch. But what sold it for me was the story. Every other FPS I had seen basically gave you a small overview (prisoner escaping from confinement, space marine killing hellspawn) and then let you kill everything. Each level of Dark Forces was a mission, starting with retreiving the Death Star plans to infiltrating a base to learn what the Empire is developing next, etc. Everything made sense in a story context that would only get built upon more in the later installment, Jedi Knight.
Final Fantasy II (IV) - Released on the SNES, I remember going to a friend's birthday party and he had rented this game. For a couple of hours the four of us there played through, watching the storyline and seeing Cecil go from loyal Dark Knight to questioning his actions. The others began to get bored and went off to do other things, but I kept playing. Hours later my friends came in and were like, "You're STILL playing this?" I played it for hours more. I thought I was almost finished by the time I finally put it down, though later once I got my own copy I found out how wrong I was. Seeing this story unfold was amazing. I had played the original FF a little bit, as one of my friends owned it, but the difference between the first and the (what I thought at the time) second was night and day. The first gives you an overall goal, but very little direction. The characters you control have no personality whatsoever, no voice, no say in anything. You just follow instructions. II on the other hand gives every character a voice, a personality, a reason they are involved in this quest. And, of course, it gave us the line "you spoony bard," which shall live forever. FF2/4 showed me what a video game could be capable of in terms of story and sheer scope.
Metal Gear Solid - this game basically taught me what I should expect from games of the generation. A great (if somewhat crazy) story, good cutscenes, tense action, lots of options in how you approached each new area, MGS was all around a great game. It is one of the first games I played where I could actually feel sorry for the villians. There was just something tragic about most of them, especially Vulcan Raven and Sniper Wolf, that even though I needed to defeat them, I still felt kinda bad about it. Overall it was just one of the most original experiences in video gaming.
Among a ton of other games, I would say the above 5 have influenced me most as a gamer.