The word fairy derives from Middle English faierie (also fayerye, feirie, fairie), a direct borrowing from Old French faerie (Modern French féerie) meaning the land, realm, or characteristic activity (i.e. enchantment) of the legendary people of folklore and romance called (in Old French) faie or fee (Modern French fée). This derived ultimately from Late Latin fata (one of the personified Fates, hence a guardian or tutelary spirit, hence a spirit in general); cf. Italian fata, Portuguese fada, Spanish hada of the same origin.
Fata, although it became a feminine noun in the Romance languages, was originally the neuter plural ("the Fates") of fatum, past participle of the verb fari to speak, hence "thing spoken, decision, decree" or "prophetic declaration, prediction", hence "destiny, fate". It was used as the equivalent of the Greek Μοῖραι Moirai, the personified Fates who determined the course and ending of human life.
To the word faie was added the suffix -erie (Modern English -(e)ry), used to express either a place where something is found (fishery, heronry, nunnery) or a trade or typical activity engaged in by a person (cookery, midwifery,...