Street Fighter II (Arcade, SNES, Genesis etc.) - Mortal Kombat can go suck itself. Street fighter is THE fighting game. Kombat was developed later because Street Fighter II was so good. Everythign about this game made fighters what they are today (for the good and the bad). Not only is it one of the most influential games ever, it holds up today still being ported and sold through various incarnations of itself and multiple sequels.
The Legend of Zelda (NES) - First game to use memory in a cartridge, had an open non-linear gameplay, revolutionized gaming.
Dune II (PC) - First RTS, while not popular, and pretty ****ty overall, it is the game that defined a RTS, leading for Warcraft and later Starcraft to take massive hints on game design.
Goldeneye (N64) - While it isn't the first FPS, nor is it the first Multiplayer FPs,Gameplay design made it one of the bes tand most influential FPS since Doom rocked PCs a few years before. The Single player was fantastic, not just a kill everythign till you get to the end. It required missions, stealth time all the things you see in major FPS titles.
For a console and four gamer slots it defined the console FPS, back when everyone else was ass backwards about multilayer. It also had free aiming, sniping, mines, etc. All the viable options that would come much later for most FPS.
Tetris (gameboy) - Nintendo is alive right now as the leading handheld gamine company because of this game. Nintendo landing the rights was a very strategic move, and since then the Gameboy brand has endured to be the number one selling gaming hardware ever. Not only that it was just pure fun, it required quick reflexes, quick thinking strategic planning etc.
Metal Gear Solid (Playstation) - What is more to say about metal gear solid? It was highly influential and all around a solid experience. For a playstation game, and a whopping 4 disks, It boasted amazing graphics, phenomenal Music, and incredible Voice Acting. The time period was not know for having such good voice actors, just see a Resident Evil Title at the time (actually see any Resident evil title, time has not sweetend capcoms ability to hire good actors and dialogue writers.) It was a griping adventure, with twists and turns, and you never quite know what is going on until the balls to the wall amazing finale.
This game was hard to divorce the gameplay from the story. It defined the cinematic experience of a video game. Taking on the gameplay of it's predecessors it was the best game on the Playstation, and spawned a number of ****ty imitators and bad sequels. While they haven't manged to make another game quite as grand as the Metal Gear Solid experience, we still have this game to play.
Diablo (PC) - This is action RPG. There is no bull****, this teh genre to the core. While better games have come out since, Diablo stands as not only the definitive action RPG title, but also the game (along with Warcraft II) to launch battle.net an unprecedented gaming service. It is still very much playable on Battle.net, although this was pre-exploit fixes, and hacking detection. Regardless the fact is Diablo was free, in a sea of online games who all sided with fees to play. This is why Diablo was so amazing, along with it's disturbing story, frightening visuals, and gory gameplay. all in all a must play for anyone who doesn't know where these games came from.
Super Mario 64 (Nintendo 64) - with the Playstation 3D gaming was out of the gate first, but stumbled. Until Mario 64 there was no good completely 3D adventure quite like it. With the Nintendo 64s 360 degress analog stick, Control of Mario had never been so good. Controler options aside, the game itself was spectacular. You could choose where you wanted go, what stars to collect, when to beat the game, or just mess around in your favorite levels. IT had some of the most memorable locations, characters, and experience I guarantee anyone from that era still remembers. Remade for the DS, and one of the 64's best selling game, it stands the test of time as one of the greatest games ever made.