mid-15c., Achademie, "the classical Academy," properly the name of the public garden where Plato taught his school, from Old French (Modern French Académie) and directly from Latin Academia, from Greek Akademeia "The Academy; the grove of Akademos," a legendary Athenian of the Trojan War tales (his name, Latinized as Academus) apparently means "of a silent district"), original estate-holder of the site.
The A[cademy], the Garden, the Lyceum, the Porch, the Tub, are names used for the five chief schools of Greek philosophy, their founders, adherents, & doctrines: the A., Plato, the Platonists & Platonism; the Garden, Epicurus, the Epicureans, & Epicureanism; the Lyceum, Aristotle, the Aristotelians, & Aristotelianism; the Porch, Zeno, the Stoics, & Stoicism; the Tub, Antisthenes, the Cynics, & Cynicism. [Fowler]
Compare lyceum. By 1540s the word in English was being used for any school or training place for arts and sciences or higher learning. "In the 18th century it was frequently adopted by schools run by dissenters, and the name is often found attached to the public schools in Scotland and Northern Ireland" [Encyclopaedia Britannica," 1941]; hence, in the U.S., a school ranking between an elementary school and a university. "In England the word has been abused, and is now in discredit in this sense" [OED]. By 1560s it was used for "a place of training" in any sense (riding schools, army colleges).
The word also was used of associations of adepts for the cultivation and promotion of some science or art, whether founded by governments, royalty, or private individuals. Hence Academy award (1939), so called for their distributor, the U.S.-based Academy of Motion Picture Arts and Sciences (founded 1927).