Battle Mastery is a Tag-Word Ability that was added in, and is exclusive to, the third Bakugan Battle Planet card set, Age of Aurelus. It only exists on Aurelus cards. It simply is a fancy way of saying choose one.
Battle Mastery cards have two effects, and the user must select one effect to use. This is also clarified after every instance of "Battle Mastery" by adding "Choose one" in every card effect.