The document discusses the branches of the external carotid artery, including the medial group, terminal branches, and maxillary artery. The medial group consists only of the ascending pharyngeal artery, which supplies structures like the pharynx. The terminal branches include the superficial temporal artery and its frontal and parietal branches. The maxillary artery travels to the infratemporal and pterygopalatine fossae, giving off branches like the inferior alveolar artery to structures of the face and lower jaw.