What is the name for bullfighting in Spanish?

A torero

Matador, The style and bravery of the matador is regarded as being at least as important as whether or not he actually kills the bull.

A picador is a bullfighter who uses a lance while on horseback to test the bull's strength.

banderillero is a torero who sets the banderillas (lit. little flags). These are colorful sticks with a barbed point which are placed in the top of the bull's shoulder.

A bullfighter is a torero. A torero who kills bulls is the matador. All matadores are toreros but not all toreros are matadores.