These type of games are characterized by an emphasis on military/police realism, heavy use of tactics and punishing difficulty.
The "official" father of the genre is Rainbow Six, but there were attempts to re-create more realistic-feel in FPS games before that. (e.g. GoldenEye 007, NAM, ect.)
The genre was the strongest on PC but consoles also got several ports. (for example Ghost Recon 2 was a console-exclusive - never played it myself tho)
Anyway, for me, Operation Flashpoint was the best of them all.
At the time, it was trully amazing to play a "real" military war game with single-shot deaths (including yourself), smart AI, usable (combat) vehicles and a huge open world without any loading during gameplay.
Games like Delta Force, (poor but interesting) Wargasm and BattleZone paved the way, but the concept of a fully open-world military game was not achieved untill Operation Flashpoint came out. (the game later spawned the "Arma" series - the current-gen OFP titles are off-shots and not connected to the original OFP in other than name)
Did you play any yourself and which did you enjoy the most?