"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')
n.
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.
The order of mammals that includes monkeys, apes, and human beings. Primates are distinguished from other animals in that they generally possess limbs capable of performing a variety of functions, hands and feet adapted for grasping (including opposable thumbs), flattened snouts, and other anatomical features. (See Linnean classification.)