Team Deathmatch, commonly abbreviated TDM is the basic team game mode of Call of Duty. It appears in all major Call of Duty games except Call of Duty 3, where it is called Team Battle. The premise is exactly the same though, two teams fight [earning points each kill] to gain the most points.
In the Hardcore variations of these maps, players may kill other players on the team. The damage will then be reflected on the player, generally killing them. Repeating this too many times will get the player banned from the lobby.