The two teams fighting over a control point on cp_freight
Gamemodes
This is a list of game modes in Team Fortress 2. Initially, the game shipped with only three game modes: Capture the Flag, Control Point, and Territorial Control. Numerous updates have added seven new game modes (Payload, Arena, King of the Hill, Payload Race, Medieval Mode, Special Delivery, and Mann vs. Machine) making ten official game modes, along with two training maps. These game modes are spread across 53 official maps and offer a great variety of play styles and games in Team Fortress 2.
Arena
Arena maps are designed to keep the class diversity of Team Fortress 2 while focusing goals around combat between two teams. Whereas other game modes lean towards a broad, overall strategy for the team, Arena concentrates on the specific tactical choices the teams make in a single fight. Arena maps carry the arena_ prefix.
Capture the Flag
Capture the Flag features both RED and BLU bases, each containing its own Intelligence briefcase (also known as "The flag" or "The Intelligence"). The goal for both teams is to capture the enemy's Intelligence until they have reached the game's capture limit (default limit is 3 captures) while preventing their enemy from achieving the same. Capture the Flag maps carry the ctf_ prefix.
Control Point
Control Point (CP) is a "main" game mode, consisting of 4 of the original 6 maps. The game revolves around capturing and defending various control points. The team that is able to take control of all the control points in the map wins. Control point maps carry the cp_ prefix.
King of the Hill
King of the Hill is similar to Arena. King of the Hill focuses on a single control point at the center of the map, which is neutral and locked at the beginning of the round. Teams must make their way to the control point and capture it when it becomes available. Once the point is captured by a team, their team clock will start a three minute countdown. If the enemy team manages to capture the point back, their clock will start counting down while the other team’s clock freezes at the time the point was recaptured. A team wins once they own the point and their three minutes is expired. King of the Hill maps carry the koth_ prefix.
Mann vs. Machine
Mann vs. Machine is a Game Mode that was released in the update of the same name. In this game mode, a team of six players must fend off a horde of robots attempting to carry a bomb to one of Mann Co.'s strongholds. Upgrades and Canteen power-ups can be purchased at the spawn point using cash picked up by the players upon destroying a robot (not every robot drops cash). A bonus amount of cash will be awarded to the team if all the cash from a wave is picked up. Mann vs Machine maps carry the mvm_ prefix.
Payload
In Payload maps, BLU team must escort a cart full of explosives through a series of checkpoints and into RED's base within a certain amount of time. BLU team members move or 'push' the cart by standing next to it - the more people next to the cart, the faster it moves (up to a maximum capture rate of three). Any RED team member standing near the cart will stop it from venturing further regardless of the number of BLU team members on the cart. Payload maps carry the pl_ prefix.
Special Delivery
Special Delivery is a Game Mode released as part of the Pyromania Update. In this game mode, an initially neutral briefcase of Australium must be transported to Poopy Joe's rocket before the opposing team can do the same. The player who has a hold of the Australium must stand on the platform, which will be raised to the compartment located at the nose cone of the rocket. The player has to remain on the platform until the compartment hatch fully opens and is ready to receive the Australium. Once the Australium is deposited into the compartment, the rocket launches off and the team wins the round. Special Delivery maps carry the sd_ prefix.
Territorial Control
In Territorial Control, the goal is to take over the entire map by capturing "territories". Each game is randomly selected from the six available layouts in a 'point against point' game where both teams must capture the opposite point while defending their own. After a team successfully captures the opposite point, the next round takes place in a different area of the map which is also randomly selected. After a team captures all four territories, that team must capture the enemy team's base. If the base point is captured the attacking team wins the game. Territorial Control maps carry the tc_ prefix.
Training
Training maps are intended to teach the basics of gameplay and advanced techniques on wooden targets and bots. Training Mode shipped with the Mac Update, which introduced two official training maps. The Offline Practice allows for new and unfamiliar players to train on specific game modes, along with the capability to adjust the difficulty scale of bots to suit the player. Training maps carry the tr_ prefix.
Other Gamemodes
Community game mode mods are custom, unofficial game modes, produced by talented community members. They vary from remakes of Team Fortress Classic game modes to completely original modes, and may change anything about the way a class works from their core mechanics to the weapons they carry. Some modes even add new enemies for the RED and BLU teams to fight, or put the player in the shoes of someone or something they couldn't play in an official mode. Some game mode mods have been honorably mentioned by Valve on the TF2 Official Blog.