Matches need three ingredients to be effective: an igniter, a fuel and an oxidizer. In a match, these roles are played by red phosphorus, sulfur and potassium chlorate, respectively.
While there are three necessary ingredients to make a match work, there are other secondary ingredients that have been added to modern matches. For instance, ammonium phosphate is put into the matchstick wood to stifle afterglow once the flame dies. The stick is also bathed in hot paraffin wax to ensure that the flame burns down past the head of the stick, and powdered glass inside the head provides increased friction to aid in striking.