The maxillary artery arises from the external carotid artery and divides into three parts - the mandibular, pterygoid, and pterygopalatine parts. It supplies structures in the face and nasal cavity. The pterygoid plexus of vessels anastomoses with the facial vein and cavernous sinus, allowing for spread of infections. Injuries to branches of the maxillary artery can cause nosebleeds, epidural hematomas, or complications from surgery like Le Fort I osteotomies.