Space Invaders (Arcade, 1978)

You've heard me talk about the "difficulty curve" in video games before.  It's the term for the natural increase in a game's difficulty as you progress through it.  Any good game tries to keep this at a steady pace, so not to make early parts of the game too hard, or later parts of the game too easy.  Space Invaders (1978) is the game that invented the difficulty curve, and by complete accident.  When programming the game, the developers made a mistake that caused the aliens on the screen to move faster as their numbers slowly dwindled away.  Voila, the first game that gets harder as you get better.
The gameplay of Space Invaders is so ingrained in gaming's subconscious that I probably don't need to explain it, but being one of the first of its kind, it sticks around in our memories.  You control the green spaceship in the never ending mission to defend Earth from the eponymous aliens.  There are four barricades (floating in... space?) which you can hide behind to shield yourself from enemy fire, but they break away with each shot they endure.  As stated, the more aliens you dispel, the faster they get, until there is only one extra-terrestrial zooming across the screen.
The largest lasting legacy of Space Invaders is how many other games of this kind it inspired.  Midway's Galaga, Konami's Gradius, and even Nintendo's Star Fox owe their success to the groundwork laid by this game.  It received a port to the Atari 2600, and it is a nearly identical port to the arcade, but with the addition of several dozen gameplay variations.  You can play with the barricades moving, no barricades, with another player, or have the aliens invisible.  Space Invaders would have a plethora of imitators on the 2600, and for years to come elsewhere, and to be honest, some of those later games are better, but would not have existed if it weren't for this early gem.