BP Potion is an item of consumption.
This item regenerates Full BP for Mirror Battle, it can be used when your BP isn't at his limit. If you use this potion whereas you still have BP, it will regenerates until the limit and your previous BP will be wasted.
It can be a reward for Login Bonus and Missions.
It's recommended to buy the packs containing tickets in the Magia Stone Shop, and when starting out the packs containing only potions are also worthwhile.