For most important or influential, I'd definitely list games like Pong, Space Invaders, Pac-Man, Super Mario Bros., The Legend of Zelda, Ultima, Street Fighter II, Doom, Quake, and Super Mario 64.
For greatest, I'm going to be boring and pick The Legend of Zelda: Ocarina of Time. And, if you'll allow me, I'll just copy what I wrote for one of the greatest games events:
Ocarina of Time is a masterpiece, the pinnacle of Nintendo creativity and ingenuity, and the benchmark against which all games -- past, present, and future -- must be judged. It pioneered innovations like z-targeting, auto-jump, and context sensitive buttons. It took existing game mechanics like horseback riding, fishing, and the realistic passage of time and wove them into the game narrative like never before. Truly, Ocarina has everything: a sweeping musical score for the ages; vast, fully-realized three-dimensional worlds; innovative and easy-to-use controls; an epic and emotionally engaging storyline; and gameplay so rich, varied, and engaging that it has no equal.
Great thread :)