Tap 'Fight' followed by 'Versus' to access the different arenas you can access. These include amongst others Avengers' Tower, Asgard Throne Room, Dr. Strange Sanctorum, and Hell's Kitchen. Arenas that expire will be replaced replaced by another character arena.
Single Match Arena (Tiers 1-4)
These never expire, just select any character and fight.
Character Arena (Tiers 2, 3-4)
These expire after 4 days. Use all your characters to rank on the leaderboard and get prizes.
Catalyst Arena (Tiers1-4)
These expire after 4 days and are the same as character arena except the rewards are rare catalysts. You must pay a fee of 100 Units to get in one.
Crystal Arena (Tiers 1-4)
These expire after 24 hours and are the same as Catalyst arenas except the rewards are valuable crystals. You must pay a fee of 2,500 Battle Chips to get in one.