Battle: These challenges involve using various weapons, doing damage and defeating your opponents in various ways.
Endurance: These challenges require you to reach gameplay milestones like leveling up or getting a certain number of wins.
Skill: These challenges require you to pull off difficult shots and really push how you play the game.
Crazy: These challenges are borderline impossible but come with substantial rewards.
Each one requires you to fully exploit the mechanics of the game and get into just the right situation to accomplish them.
Completing challenges unlocks medals which can be used to boost your stats and grant special abilities in game without having to change your equipment load-out.