Bullet Arts is a fighting style utilized by the Umbra Witches which uses a combination of melee attacks and gunfire, particularly from guns used on both hands and feet. It is used by both Bayonetta and Jeanne throughout the game. It allows the fighter to execute strings of fast paced moves as well as combine them with the use of Wicked Weaves.
The Bullet Arts also allows the fighter to become efficient at any weapon they happen to come across using the same configuration of hands and feet, explaining Bayonetta's efficiency in being able to execute various attacks with different weapons as well as utilize the Angel Arms she can pick up.
The Arts appear to be exclusive to the Umbra Witches, as the other members of the clan are also seen to utilize both melee and gunfire in the flashbacks against enemies.