If someone is “putting on airs” it means that he or she is acting superior or snobbish.
Since the 1500s, “airs” has referred to having an affected manner. It’s from the French word air, “look, appearance, or bearing.” Behaving as if you’re better than other people — wealthier, better dressed, or better educated — is to put on airs. Acting like you know more than your teacher is a way to put on airs.