An injection of anesthetic at the actual site of surgery is called local anesthesia. It is usually only useful for operations on small areas. This is sometimes used with monitored anesthesia care (MAC).
Monitored Anesthesia Care (MAC)
In this case, the surgeon uses local anesthetic while the anesthesiologist provides sedation or "light anesthesia." The anesthesiologist both fine-tunes the sedation and manages the patient's medical conditions. Sometimes a trained anesthesiologist is asked solely to help manage medical conditions.