Introduction: this vessel begins at the bifurcation of the
common carotid, at the level of the superior border of the thyroid cartilage
Course: it ascends beneath the twelfth cranial nerve and the
posterior belly of digastrics to enter the parotid gland. In the parotid gland
it lies deep to the fascial nerve and the retromandibular vein.
Termination:  the
external carotid artery ends within the parotid gland at the level of the neck
of the mandible by dividing into the superficial temporal and internal
maxillary artery.
Relations: at first it is deep to the internal carotid
artery, then passes anterior and lateral to it.
The internal 
jugular vein is first lateral and then posterior to it.
Medially lies the pharynx.
Within the parotid gland the carotid artery lies
deep to the fascial nerve and the retromandibular vein.
Branches: before entering the parotid gland the external
carotid artery gives off six branches: three from the anterior side, two from
the posterior side and one medial.
Anterior side:
Superior thyroid artery.
Lingual artery
Fascial artery
Posterior side:
Occipital artery
Posterior auricular artery.
Medial side:
Ascending pharyngeal artery.
Terminal branches: these are formed within the parotid gland
and they are:
Superficial temporal artery
Internal maxillary artery.