N64/PS1/Quake/Duke 64 is still best era for first person shooters, don't @ me
Late 90's PC - Unreal Tournament, Half Life, Quake 3 Arena. And Deus Ex if you could consider it.
Also to a lesser extent, the late 2000's PC/X360 for innovating with the genre - Mirror's Edge, Portal.
90's FPS games were groundbreaking for sure. I'd argue that 2004-2007 was when the genre peaked. Half Life 2 was adored in 2004, Halo 2 online multiplayer laid out the groundwork for matchmaking and features for console shooters to this day.
2007 had Bioshock, Halo 3, Team Fortress 2, Portal (not a shooter but was adored), Call of Duty 4, HL episode 2. Then a but later there was Left 4 Dead and Crysis.
FPS games haven't really moved past framework set in the 360/ps3 generation. Call of Duty is to blame for that.