I'd say there are a few that I've counted as best over the years. Bear in mind that this list represents my views at the time the game was current. A modern game like Halo Reach I feel is better than the ancient predecessor of Halo for example but having a list like this is boring. So, in rough chronological order:
Doom - it was my first FPS and was thus the best one I'd played. It also remains one of my most played games of all time as I always seem to purchase a new version of the game for almost every platform that runs it.
Duke Nukem 3d - There were a lot of games between Doom and Duke but Duke was simply better in every way. Monologs, highly interactive environments, interesting weapons and staggering technology made it a winner. I mean, brass was ejected from guns and the left bullet holes in stuff and you could blow holes in walls! No game did all of that (or even most of that) at the time.
Goldeneye - The best FPS made for consoles to date and the greatest party game of the era. Perfect Dark was better in all respects but not by enough to make this list using the criteria above.
Team Fortress (Mega) - to this day the influence of TF cannot be overstated. It popularized objective based games. It was the first notable game to go class based and give players weapons from the start. I cannot think of a multiplayer FPS made in the last decade that was not influenced by this one.
Unreal - Setting and world design. This game was a revelation as FPS game tended to be a series of disconnected mazes and Unreal demonstrated that something that is largely linear is probably better.
Half-Life - Better than Unreal in important areas. Unreal was the better shooter. Half-Life did a better job of building a world.
Tribes - My favorite pure FPS multiplayer game of all time. Nothing else has come close. People who didn't go up with multiplayer in the late 90's would find it baffling, but Tribes was largely played by people with dial up internet connections and there was no lag compensation system. You input was delayed (in my case generally by 2/5 of a second) and there were few hitscan weapons (most projectiles took awhile to get where they were going). In frantic battles in the right terrain, one could engage in combat at hundreds of miles an hour.
Halo - The best console FPS to date (again). A revelation that the console could play a proper FPS in spite of the lack of keyboard and mouse. So much about this game is still seen today. The obligatory vehicle sequences in the modern FPS are generally worse versions of the Warthog/Scorpion levels. Mechanics like regenerating health were popularized by this game. I know it wasn't the first on that last point, but it's success was such that the design philosophy shifted to the extent that the maze exploration health meter FPS is now the rare exception. How influential was Halo? Enough that just about every FPS game made since has largely copied the control scheme.
Battlefield Bad Company 2 - The best execution of the Battlefield formula. Better map design ensured that no side was particularly favored in a rush map (most of the time) and rush itself simply offers a better game for the most part. Conquest all too often becomes a game of whack-a-mole. Sure, exceptions existed. Wake Island remains one of the great multiplayer spaces. In bringing relative balance to the sides, it finished what Bad Company began by making the game more fun to play. I'm sure there are those who remember just how horrifically lethal the old battlefields were on foot. With Bad Company, staying alive behind slabs of armor was just as hard as doing it on foot.
