Cryosurgery and Electrocautery are the two methods used for removing moles and other skin lesions, including warts and skin tags.
Cryosurgery is controlled destruction of surface tissue lesions (or light moles) by rapid freezing. This method is used mostly on the face because it has very minimal scarring.
Electrocautery is done with a machine called a hyfrecator which removes the lesion by a carefully delivered spark of electricity. Of course, local anesthesia is used for these procedures whenever possible.