The modern day FPS is the son of one game: DOOM. Published by the now-legendary id Software, DOOM broke the mold. It wasn't the first FPS, but it redefined the genre with better gameplay, some excellent multiplayer, and of course a shotgun, a BFG and a chainsaw.
Playing DOOM now isn't nearly as immersive as playing a modern horror FPS, but at the time DOOM was quite freaky. Going up against demons with a shotgun and a chainsaw is harder than Ash makes it look, believe me.
DOOM was cool to play multiplayer. It was the first FPS I ever saw played that way, even if I never got the chance to play myself. I was watching [a friend's dad] play against a friend over a phone line, and the operator kept calling and asking if they wanted to keep playing. They were hosing down demons and each other with shotguns and machine guns. To me, it was magical.
The original DOOM's entertainment days are now over, and it hangs on based only on gamers' nostalgia. But fear not: the new DOOM is on its way, and it's poised to once again take over the video game world.