Old English swimman "to move in or on the water, float" (class III strong verb; past tense swamm, past participle swummen), from Proto-Germanic *swimjan (source also of Old Saxon and Old High German swimman, Old Norse svimma, Dutch zwemmen, German schwimmen), from PIE root *swem- "to be in motion."
The root is sometimes said to be restricted to Germanic, but according to OED possible cognates are Welsh chwyf "motion," Old Irish do-sennaim "I hunt," Lithuanian sundyti "to chase." The more common Indo-European word is *sna-. Transitive sense of "cross by swimming" is from 1590s. Sense of "reel or move unsteadily" first recorded 1670s; of the head or brain, from 1702. Figurative phrase sink or swim is attested from mid-15c., in early use often with reference to ordeals of suspected witches.
1540s, "the clear part of any liquid" (above the sediment), from swim (v.). Meaning "part of a river or stream frequented by fish" (and hence fishermen) is from 1828, and is probably the source of the figurative meaning "the current of the latest affairs or events" (as in in the swim "on the inside, involved with current events," 1869). Meaning "act of swimming" is from 1764.