A topical or subcutaneous chemical administered directly to the part of the body receiving surgery to eliminate pain and pressure sensations in the affected area while keeping the patient conscious during the procedure.
Several different chemicals are used, depending on both availability and suitability. These chemicals include benzocaine, procaine, cocaine (hey, don't knock it -- it works), xylocaine, lidocaine, and plenty more.
Generally, local anesthetic is much preferred over general anesthetic (which knocks the patient unconscious entirely and invites complications), as local anesthetics leave the patient conscious, usually coherent, and able to cooperate with physicians throughout the procedure. It also takes less time to wear off, involves fewer complications (general anesthesia is risky, and can result in coma, brain damage, or infection if administered improperly, or if the patient has an allergic reaction to it), and means the patient can be out the door more quickly. This is an essential requirement for outpatient surgery.
Thanks to velocipenguin for whacking me with the clue-by-four -- nitrous oxide, while administered as a gas via the mouth and nose, isn't a local anesthetic; it does things to the whole body. Thanks to BlueDragon for pointing out it's "subcutaneous", not "intravenous", for local anesthetics. Heh. Are well all glad I'm a programmer and not a physician, or what? :)