1590s, "a sending abroad" (as an agent), 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'."
Meaning "an organized effort for the spread of religion or for enlightenment of a community" is by 1640s; that of "a missionary post or station" is by 1769. The diplomatic sense of "body of persons sent to a foreign land on commercial or political business" is from 1620s; in American English, sometimes "a foreign legation or embassy, the office of a foreign envoy" (1805).
General sense of "that for which one is sent or commissioned" is from 1670s; meaning "that for which a person or thing is destined" (as in man on a mission, one's mission in life) is by 1805. Meaning "dispatch of an aircraft on a military operation" (by 1929, American English) was extended to spacecraft flights (1962), hence, mission control "team on the ground responsible for directing a spacecraft and its crew" (1964). As a style of furniture, said to be imitative of furniture in the buildings of original Spanish missions to western North America, it is attested from 1900.