From The TimeSplitters Wiki

Get down! -to work! Challenge has been given the Improvement Badge, Please contribute to this article!

The Challenges are similar to Arcade League, except that these challenges are usually have a mini-game feel to them and the objectives are often more absurd than what you would find in League.
Completing these challanges not only unlocks more challanges but additional characters and cheats for use in both arcade and single player.