1550s, "act of setting off" (on a journey, etc.), from off + set (adj.). Meaning "something 'set off' against something else, a counterbalance" is from 1769; the verb in this sense is from 1792. As a type of printing, in which the inked impression is first made on a rubber roller then transferred to paper, it is recorded from 1906.
A shoot that develops laterally at the base of a plant, often rooting to form a new plant. Many succulents and cacti are propagated by removing offsets and planting them elsewhere. See more at vegetative reproduction.