early 14c., "to bear or convey, take along or transport," from Anglo-French carier "to transport in a vehicle" or Old North French carrier "to cart, carry" (Modern French charrier), from Gallo-Roman *carrizare, from Late Latin carricare, from Latin carrum originally "two-wheeled Celtic war chariot," from Gaulish (Celtic) karros, from PIE *krsos, from root *kers- "to run."
Meaning "take by force, gain by effort" is from 1580s. Sense of "gain victory, bear to a successful conclusion" is from 1610s; specifically in reference to elections from 1848, American English. Meaning "to conduct, manage" (often with an indefinite it) is from 1580s. Meaning "bear up and support" is from 1560s. Commercial sense of "keep in stock" is from 1848. In reference to mathematical operations from 1798. Of sound, "to be heard at a distance" by 1858.
To carry out "conduct to completion" is from c. 1600. To carry it off "brazen a thing out" is from 1704; carried off as a euphemism for "killed" is from 1670s. To be carried (away) in the figurative sense "transported, having the attention fully absorbed" is from 1560s. Carrying capacity is attested from 1836. Carry-castle (1590s) was an old descriptive term for an elephant.
by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.
Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.