Battle Points, or BP, are what you receive from successfully attacking and executing players in PvP fights.
The number of Battle Points awarded for a win depends on the Battle Rank of the defeated opponent. Defeating an opponent at the same rank earns you 10 BP. The reward is increased by 2 BP per rank your opponent is above you, and likewise is decreased by 2 per rank your opponent is below you. Defeating a player 5 or more ranks below you yields no Battle Points.
After a certain amount of BP is collected, your rank will increase. There are 24 Battle Ranks in total.