"high bishop," c.1200, from Old French primat and directly from Medieval Latin primatem (nominative primas) "church primate," noun use of Late Latin adjective primas "of the first rank, chief, principal," from primus "first" (see prime (adj.)).
Meaning "animal of the biological order including monkeys and humans" is attested from 1876, from Modern Latin Primates (Linnæus), from plural of Latin primas; so called from supposedly being the "highest" order of mammals (originally also including bats).
primate pri·mate (prī'māt')
A mammal of the order Primates, which includes the anthropoids and prosimians, characterized by refined development of the hands and feet, a shortened snout, and a large brain.
Any of various mammals of the order Primates, having a highly developed brain, eyes facing forward, a shortened nose and muzzle, and opposable thumbs. Primates usually live in groups with complex social systems, and their high intelligence allows them to adapt their behavior successfully to different environments. Lemurs, monkeys, apes, and humans are primates.