From The TimeSplitters Wiki

TimeSplitters features many different modes of play. Each mode is included under a primary game-type. Story allows the player to follow the story of humanity's war with the TimeSplitters, a time-traveling alien race. Arcade League and Challenge are objective-based missions that unlock a majority of the game's features. Arcade Custom and Network are the primary means of multiplayer action. Arcade League and Network are unique to TS2 and TSFP.

Game Types

TimeSplitters Story Arcade Custom Challenge MapMaker

TimeSplitters 2 Story Arcade Custom Network Arcade League Challenge MapMaker

TimeSplitters: Future Perfect Story Arcade Custom Network Arcade League Challenge MapMaker