The arena is where battles happen(Mechatars love to show of their skills.) Every day your Mechatar is supposed to do three battles a day to keep healthy called Daily Battles. Once those are finished your Mechatar can keep on Battling(but you will earn less xp and credits.)
Your Mechatar can also do training battles. They don't go into your record like official battles but are used to help your Mechatar get in shape. In order to do them you have to make allies.
Official battles are stored in here. Its main use is for recording battles however it can also be used to create allies by clicking the group,clicking the magnifiying glass and click ally.