The maxillary artery has three parts: 
🟥 Mandibular section: Supplies the jawbone, lip and chin area, and middle ear
🟥 Pterygoid section: Supplies the muscles used for chewing
🟥 Pterygopalatine section: Supplies the roof of the mouth, jaw, teeth, lips, eyelids, and inner ears
BRANCHES OF THIRD PART OF THE MAXILLARY ARTERY 
The third part of maxillary artery, is anterior to the lateral pterygoid muscle. 
It passes from between the two heads of lateral pterygoid muscle to enter pterygopalatine fossa : 
Mnemonic: [PIGS-PP]
🟥 POSTERIOR SUPERIOR ALVEOLAR ARTERY: 
The posterior superior alveolar artery arises just before the maxillary artery enters the pterygomaxillary fissure. Descending on the posterior surface of the maxilla it gives off branches that enter canals in the bone to supply the molar and premolar teeth, and the maxillary sinus. 
🟥 INFRAORBITAL ARTERY: 
The infraorbital artery also arises just before the maxillary artery enters the pterygomaxillary fissure. Through the inferior orbital fissure it enters the orbit and runs forwards in relation to the floor of the orbit, first in the infraorbital groove and then in the infraorbital canal emerging on the face through the infraorbital foramen. It gives off some orbital branches to the structures in the orbit and the anterior superior alveolar branches to the incisor and canine teeth. After emerging on the face, the infraorbital artery gives branches to the lacrimal sac, the nose and the upper lip. 
🟥 GREATER PALATINE ARTERY: 
The grater palatine artery runs downwards in the greater palatine canal to emerge on the posterolateral part of the hard palate through the greater palatine foramen. It then runs forwards near the lateral margin of the palate to reach the incisive canal through which some terminal branches enter the nasal cavity. Branches of the artery supply the palate and gums. Within the greater palatine canal it gives off the lesser palatine arteries that emerge on the palate through the lesser palatine foramina and run backwards supplying the soft palate and tonsil. 
🟥 SPHENOPALATINE ARTERY: 
The sphenopalatine artery passes medially through the sphenopalatine foramen to enter the cavity of the nose. It gives off posterolateral nasal branches to the lateral wall of the nose and to the paranasal air sinuses, and posterior septal branches to the nasal septum. 
🟥 PHARYNGEAL ARTERY:
The pharyngeal branch runs backwards through pharyngeal or palatovaginal canal. It supplies part of the nasopharynx, the auditory tube and the sphenoidal air sinus. 
🟥 PTERYGOID CANAL ARTERY:
The artery of the pterygoid canal runs backwards in the pterygoid canal and supplies the pharynx, the auditory tube and the tympanic cavity.