Local anesthesia and General anesthesia are the two types of anesthesia. Anesthesia is process of using medication to cause a state in a person in which the person undergoing any surgical operation does not feel any pain. This is possible because pain sensation will be killed in the area where surgical procedure is taking place or the person is made to fall into a deep sleep.
Local anesthesia, when applied on a person causes the person not to feel pains at the specific place the surgical operation is going to take place. While general anesthesia causes someone to fall into a deep sleep so that no pain is felt during the medical treatment. In local anesthesia you are aware that a medical treatment is taking place on you, just that you won't feel it, but in general anesthesia you won't even know that you have been operated until you wake up.