Anesthesia is the medicine or gases given to the patient to make him/her sleep or get pain free. Commonly used anesthesia types are Local Anesthesia and General Anesthesia.
Some procedures are non invasive or minimally invasive, like botox and fillers and lasers. So no anesthesia is given to the patient. However sometimes medicines are given to take orally and patient feels absolutely no pain.
Local Anesthesia means that doctor will inject a medicine called anesthetic to the patient and the area of procedure will become numb. Patient will be waking during the operation and can talk. After procedure patient can immediately start walking as nothing has happened ever and go home.
It means that patient will be injected with an anesthetic or given some gases by mask. Patient will imediately sleep and will wake up when procedure is already done. Patient will be unaware of all events like nothing has happened. Patient can go home when safe.