1520s, "make a spring from the ground" (intransitive), a word with no apparent source in Old or Middle English, perhaps imitative (compare bump (v.)); another theory derives it from words in Gallo-Roman dialects of southwestern France (such as jumba "to rock, to balance, swing," yumpa "to rock") and says it might have been picked up during the Hundred Years War. Similarities have been noted to Swedish dialectal gumpa "spring, jump," German dialectal gampen "jump, hop," but OED finds no basis for a relationship.
It has superseded native leap, bound, and spring in most senses. Meaning "pass abruptly from one state to another" is from 1570s. Meaning "move suddenly with a leap" is from 1724. The transitive meaning "to attack, pounce upon" is from 1789; that of "to do the sex act with" is from 1630s. Related: Jumped; jumping.
Sense in checkers is from 1862. To jump to "obey readily" is from 1886. To jump to a conclusion is from 1704. To jump rope is from 1853; Jumping-rope (n.) is from 1805. Basketball jump-shot "shot made while the player is in the air" is from 1934; also used of billiard shots. Jump in a lake as a dismissive invitation is attested from 1912.