OK, as soon as I saw the bounty I couldn't resist.
Best FPS ever for me is Unreal Tournament (I, II, III, IV), for the following reasons:
Super fluid and efficient graphic engine: Unreal Engine, still used today btw, reached version 5
Variety of weapons: long range, medium range, melee, rockets, grenades, energy weapons with the possibility of creating combos (see plasma rifle)
Various gameplay for single and multi player: deathmatch, team deathmatch, capture the flag, assault, instagib
Possibility of using vehicles on latest versions of the game
Great maps and scenarious
Maps editor for players
Great mobility: dodging, dodging on walls, jump boosters, sliding on surfaces
Great e-sport competitions all around the world
Possibility of giving orders and commands to bot team members
Great futuristic-dystopian atmosphere