For single player, I would have to say half-life, but I am really starting to hate what it has done for fps games. It seems like pratically every fps game that comes out now is ultra-linear and ultra-scripted. How can I be immersed in a world were I know everything that happens has been pre-planned by the developer, a world where nothing I do has an effect on anything and all I can do is follow the path?
Call of duty is the worst example of this. It was really cool at first because of the atmosphere and so on but after a while it begain to feel pointless playing it because I knew none of my actions made a difference. Anyone that dies on either side is just replaced, and certain key characters are invincible anyway. All you have to do is keep yourself alive until the game advances the plot. Ironically the game that looks like it has the best chance of breaking this trend is half-life 2.
For multiplayer I'm going with Quake 1, even though I never really played it (but i played a LOT of quake 2). This was the game that really opened up competitive online gaming, which I think was a more important event than even the creation of the fps. I know I would rather live without first person shooters completely than live without online games completely. The whole Quake series remains a better test of skill than any online fps out today. They are pretty much the only games where if 2 experts play 1 on 1, the better player will concistently win. Unfortunatly, this is also what killed their popularity since they are not newb friendly.