mid-14c., originally of religion, "a radical and complete change in spirit, purpose, and direction of life away from sin and toward love of God," from Old French conversion "change, transformation, entry into religious life; way of life, behavior; dwelling, residence; sexual intercourse," from Latin conversionem (nominative conversio) "a turning round, revolving; alteration, change," noun of action from past-participle stem of convertere "to turn around; to transform," from assimilated form of com "with, together" (see con-) + vertere "to turn" (from PIE root *wer- (2) "to turn, bend").
Sense of "a change from one religion to another" (especially to Christianity) is from c. 1400 in English. General sense of "transformation, a turning or changing from one state to another" is from early 15c. In reference to the use of a building, from 1921. Conversion disorder "hysteria" (attested from 1946 but said to have been coined by Freud) was in DSM-IV (1994). Conversion therapy in reference to homosexuality is by 1979.