The descending palatine artery is one of the terminal branches of the maxillary artery, which in turn arises from the external carotid. It supplies the soft palate, the gums, and the mucous membrane and bone of the hard palate.
The descending palatine artery originates from the pterygopalatine part of the maxillary (third portion in some text). Right before it takes a downward course, it gives off its first branch, which is the artery of pterygoid canal. Then it descends into the greater palatine canal where it forks into two branches 1) the greater palatine artery and 2) the lesser palatine artery. The former supplies the hard palate, its mucous membrane and the gum, while the latter brings oxygenated blood to the soft palate and part of tonsil.
Below, schematic picture of nasal cavity and mouth, showing the descending palatine artery and its branches. The third portion of maxillary cannot be seen because it is hidden in the pterygopalatine fossa.