Old English rap, from Proto-Germanic *raipaz (cf. Old Norse reip, Middle Dutch, Dutch reep "rope," Gothic skauda-raip "shoe-lace," Old High German, German reif "ring, hoop"). Technically, only cordage above one inch in circumference and below 10 (bigger-around than that is a cable). Nautical use varies. Finnish raippa "hoop, rope, twig" is a Germanic loan-word.
To know the ropes (1840, Dana) is originally a seaman's term. Phrase on the ropes "defeated" is attested from 1924, a figurative extension from boxing. To be at the end of (one's) rope "out of resources and options" is first attested 1680s. Formerly also in many slang and extended uses related to hanging, e.g. John Roper's window "a noose," rope-ripe "deserving to be hanged," both 16c.