The battle Arena works like this: (I'll have to use an example)

Example:
Sam (HP: 24, Attack: 16, Defence: 5) vs. Fred (HP: 20, Attack: 12, Defence: 4)

Sam attacks Fred so....

Step 1) Sam's attack - Fred's defence = Sam's damage
16 - 4 = 12

Step 2) Fred's HP - Sam's damage
20 - 12 = 8 HP left for Fred

There's will be contests enabling you to upgrade your character's status.

Jewels increase HP
Shells increase Defence
Claws increase Attack

There'll be more available soon.