Taken from UDN:
Every game made with Unreal Engine revolves around the use of one or more gametypes. This concept is obvious in games such as Unreal Tournament where the user actually chooses the type of game to play, whether it is Deathmatch, Capture the Flag, Last Man Standing, or some other gametype. However, even games such as platformers, sports and racing games, RTS games, RPGs, etc all rely on a gametype, though it is transparent to the user.
A gametype is what determines the rules of the game: how the game starts, how scoring or other events are handled, what conditions must be met for the game to end, and any other elements of the game. The game and thus the gametype is also level-specific; when a new level is loaded (At least in the traditional sense. Level streaming loads levels into the current level leaving the current gametype.), a new gametype is loaded. In a Deathmatch game, the gametype handles spawning the players and bots into the game at various locations, handling kills by each player and keeping score, and ends the game when either the time runs out or one player has made the required number of kills. This is all very straightforward and the same concepts apply to any other game, although they may be slightly more complex.
For instance, imagine an adventure role-playing game. When the player starts the game, the gametype spawns the player into the world, either at a designated location, such as the opening of a dungeon or where the player was when the game was last saved. The player explores the world and then comes to a dungeon. When the player enters the dungeon, a new level and gametype are loaded. This gametype spawns the player at the beginning of the dungeon and then waits for the player to complete the objective of the dungeon, which may be killing a boss or collecting a specific item or what have you. When the gametypes detects the dungeon has been completed, it sends the player back to the main world, loading the main world level. In the main world, the gametype spawns the player at the opening of the dungeon and waits for the player to finish all the objectives of the game as they continue to explore.
Clearly this is a much more complex case as the game has multiple gametypes which all work together, but the concept of each gametype is the same as that for the simple case of a Deathmatch game.
In short, a gametype is the rules and the functionality of the game. If you're playing a deathmatch game it's in charge of spawning players, handling pickups, keeping score, and so on. It decides when the game is over, who wins, and just about everything else related to the object of the game.
It is implemented as GameInfo.