Examples of self-mutilation are common extensions of masochism. You may have purposely cut or burned yourself; you may have purposely scarred your hands, wrists, arms, or legs as a result of self-mutilation.
The movie The Piano Teacher illustrates perfectly this behavior and its psychology. Self-mutilation serves to manage overwhelming anxiety. Usually, patients report that physical pain feels much more bearable than the psychic pain that they are experiencing at the time.
The psychic pain may stem from violent urges towards significant others, or from otherwise intolerable feelings of annihilation, disintegration, or the like. Redirecting the violence toward oneself contains these urges and thus prevents one from hurting people whom one also genuinely feels that one needs.
Self-mutilation serves as a compromise between expressing the violence and containing it in a way that will not hurt others. This self-punishment can often feel soothing, as the mutilation serves to take the greater psychic pain away. It also recreates a union with the abusive parent who related to the child abusively, thus helping the patient to feel less alone, annihilated, or disintegrated.
In a less overt way, people mutilate themselves figuratively with their harsh internalized voices. Telling one-self how inadequate and horrible and nonsensical one is serves a similar purpose—keeping the pain in check, but also redirecting the rage towards oneself to protect the caretaker. These beatings can make self-mutilators feel very powerful, a genuine contrast to the sense of helplessness experienced otherwise (either from separation or another type of annihilation).