1590s, "a sending abroad," originally of Jesuits, from Latin missionem (nominative missio) "act of sending, a dispatching; a release, a setting at liberty; discharge from service, dismissal," noun of action from past participle stem of mittere "to release, let go; send, throw," which de Vaan traces to a PIE *m(e)ith- "to exchange, remove," also source of Sanskrit methete, mimetha "to become hostile, quarrel," Gothic in-maidjan "to change;" he writes, "From original 'exchange', the meaning developed to 'give, bestow' ... and 'let go, send'."
Diplomatic sense of "body of persons sent to a foreign land on commercial or political business" is from 1620s. In American English, sometimes "an embassy" (1805). Meaning "dispatch of an aircraft on a military operation" (1929, American English) later extended to spacecraft flights (1962), hence, mission control (1964). As a style of furniture, said to be imitative of furniture in the buildings of original Spanish missions to North America, it is attested from 1900.