Doom, plain and simple.
Wolfenstein 3D was the first real FPS as far as a fully fleshed out concept, and I definitely remember squealing with joy back it 1992 when I bagged my first Nazi. However, it was Doom that truly shifted FPS gaming into full gear. It was Doom made the FPS popular and solidified it as genre of gaming that was here to stay. It had multiplayer, as well as a much wider variety of enemies and weapons.
Now, in an alternate reality things might have gone differently. Doom's main competitor at the time was Marathon, and in many aspects Marathon was a superior game. Released in 1994, it was the first game to use mouse look, not Duke 3D. Also it had the ability to have one room directly on top of another. Even though you still couldn't jump it gave the player the ability to grenade or rocket hop. It also was the first to have a truly intricate story, even though you only got the story through computer terminals as cut scenes were technologically a ways off. The multiplayer was also very cool, at least on par with Doom. However, back in those days, Bungie released their games for Macintosh. PC gamers never even saw Marathon, and only after a lot of griping did Marathon 2 get ported to the PC. If you're an old time gamer like me you can also see the influence that Marathon had on the plot of Halo If you feel like checking it out the Marathon trilogy is now freeware for every OS, as well as the second game being available on XBLA. Trust me though, play it on your computer.
But because of the obscurity of Bungie back then, Doom (which I'm not trying to bash, it was great) won the popularity contest and therefore defined, and began the evolution of FPS games.