The arena has three different tabs: Arena, Pieces and Replay.
In the Arena, you can pick a fight with any of three random opponents of different difficulty (green = easy, yellow = normal, red = hard). Fighting in the Arena costs Mana and can get you Tournament Points.
In the Pieces menu, you can attack other players to steal Card Pieces that you have started collecting from them. If you win, you get that Card Piece, and the other player loses it. If you lose, you lose nothing. Fighting for Card Pieces also costs Mana and can get you Tournament Points.
The Replay tab shows your last battles and you can review them. If you were wondering about how someone stole your Card Piece, or if you want to rewatch a battle, this is the place to go!