A Team Deathmatch is when there are two teams (red and blue) on the map at the same time. Red team members are trying to kill members of the blue team, and vice versa.
Members of the same team can cooperate with each other by linking up a teammates using the link gun secondary fire (both players need to be using the link gun for this to be effective). Most games are set up so that friendly fire does not hurt players from the same team.
The goal is to end with a higher score than the other team. The first team to achieve the target score wins the match. If the timer runs out, the team with the highest score wins.
Scoring is the same as normal Deathmatch. Scoring is broken down by player in the final tally, but the total score of all players on a team is the team score.