Team Fortress itself is a series of class-based games. These classes are Soldier, Demolitions Man, Sniper, Pyro, Medic, Engineer, Spy, Scout, and Heavy.
It started out as a mod for Quake (1996) and became very popular and epic. The developers decided to start on a Team Fortress 2 for Quake II (1997), but all work on that was scrapped when they joined Valve. They then made Team Fortress Classic, a mimic of QTF on the GoldSRC engine, or the Half-Life (1998) engine.
Team Fortress 2 was again being developed, but now on the GoldSRC engine, subtitle Brotherhood of Arms. It featured realism and stuffs. When development switched to Source, TF2 was given a cartoony art style instead of the realistic one. The cartoony style has proved to be awesomely awesome and epicly epic.
TF2 is $19.99 on Steam, and TFC is $4.99
The original mod for Quake can be downloaded
Here.