Mine would have to be Return to Castle Wolfenstein. Man, the multiplayer in this game is what originally got me hooked to online gaming. I’m a much bigger fan of the fast paced FPS games like Quake 3. Don’t worry, I’m growing grey hairs.
Close second’s are
Doom
Quake 3 Arena
Unreal Tournament
Goldeneye 007
Add Half Life and CS beta to 1.3 to the list. Good old days.
My god how did I forget CS. The 1.6 days 
Doom
Quake 3
Unreal Tournament 2004
Duke Nukem
I have to admit a guilty pleasure of mine is the duke reboot. Just came out too late.
I preferred Quake 2 over Quake 3. Half-life was great. Nothing beat Unreal Tournament when it came to LAN play (Love instagib matches!)
I’m not sure if it strictly counts as a FPS but Battlefield 1942 was extremely epic, especially with the DesertCombat mod. Probably one of the best LAN Party games ever. So much so that EA games basically copied DesertCombat and turned it into Battlefield 2.
Delta Force: Land Warrior - Back when sniper scopes weren’t spotlights.
Not super old but definitely not new:
Battlefield Play4Free
Tribes: Ascend
I remember buying my first PC(a Sony Vaio) and BF1942. My goodness was that incredible to witness.
Half-Life.
Metroid Prime series.
Specifically, Metroid Prime Trilogy with the Wii nun-chuck.
I wish for a remaster of Return to Castle Wolfenstein. I enjoyed the recent New Order/Old Blood etc MachineGames series but RTCW was the best balance of weird science fiction and occult supernatural stuff.
As for the main question, growing up with Macs, I got to enjoy Marathon/Marathon2: Durandal.
I know this isn’t what people think of when they think of FPS, but it had all the elements, plus actual 3D movement:
It also had a PvP mode. Don’t tell anyone, but the IT guys and us schlubs in the Records Office had a regular mini-tournament thing going on for about a year.