All fish breathe oxygen. Oxygen is one of the ingredients required to make water. So, a fish that breathes oxygen is called a fish.
Pretty much all living things require oxygen, but maybe you are talking about fish that breathe air? There are several types that can do this, including the African lungfish, several types of catfish, and snakeheads. African lungfish require air, and will actually drown in water if they do not have access to air.
Anabantids like gouramis and bettas are also air-breathers.
A little more about the African lungfish comment: When these were first discovered, scientists wanted to bring them back to the US for study. They would put the lungfish in shallow containers large enough for the fish to swim around in with a little airspace at the top, bring the fish home and find them all dead on arrival. The obvious conclusion: African lungfish can't be transported. Only later did scientists learn the lungfish has to approach the surface of the water at an angle in order to breathe. When they started bringing the lungfish home in deep containers--deeper than the fish are long--the fish were able to survive the journey.