Meaning "individual wheeled vehicle" is c. 1400; specific sense of "horse-drawn, wheeled vehicle for hauling people" first attested 1706; extended to railway cars by 1830. Meaning "the business of transportation" is from 1520s. Meaning "way of carrying one's body" is 1590s, hence also "behavior, conduct, manners." Sense of "a part of a machinery which carries another part" is from 1680s. Carriage-house attested from 1761.
Proto-Indo-European root meaning "to run."
It forms all or part of: car; career; cargo; caricature; cark; carpenter; carriage; carrier; carry; charabanc; charette; charge; chariot; concourse; concur; concurrent; corral; corridor; corsair; courant; courier; course; currency; current; curriculum; cursive; cursor; cursory; discharge; discourse; encharge; excursion; hussar; incur; intercourse; kraal; miscarry; occur; precursor; recourse; recur; succor.
It is the hypothetical source of/evidence for its existence is provided by: Greek -khouros "running;" Latin currere "to run, move quickly;" Lithuanian karšiu, karšti "go quickly;"Old Irish and Middle Welsh carr "cart, wagon," Breton karr "chariot," Welsh carrog "torrent;" Old Norse horskr "swift."