For me I would have to say Counter-strike. Though it lacked a single player campaign, it is the most competitive online shooter still to this day. Both had outstanding graphics and gameplay from release and both are still played and well-supported making it the longest lasting active online shooter with the deepest community.
Close behind is Halo. Not only did it revolutionize console FPS, it brought about a very engaging campaign and a continually refined online experience. It is the most competitive online shooter for consoles hands down.
So what about for you what is the best FPS for you and most importantly...Why? Peace.
Edit: Obviously there are many great FPS games and I should have been more specific. It might be easier to discuss online FPS. I tried to take a franchise from each system (didn't include wii because I never played an FPS on it).
Honorable mentions for best FPS: Metroid Prime, Stalker series, Bioshock. Those are more known for single player experiences but feel free to discuss any FPS that is your favorite of all time.
I think really it comes down to Halo Vs Counter-strike when discussing best "online shooter." I should have probably just made the poll that!! Oh well.