Works the same as energy but its only used in PVP battles, or in certain events. It can be increased like energy. The amount you consume is equal to the amount of BP your formation has. Having a high BP generally shows you as a potentially tougher opponent, which may help to discourage other players from farming your gold through attacking you. The BP of your formation is solely determined by the sum of the BP's of all the monsters in your current formation, which cannot exceed your BP cap. Mainly is just for meeting the BP requirements for your formation the rewards from battling isn't as high as missions, although some battles could reward you with a much higher gold reward than from missions, but since it regenerates so fast its use now is only for battle related events.