1540s, "a going together, a coming together," from Late Latin coitionem (nominative coitio) "a coming together, a meeting; copulation," noun of action from coitus, past participle of coire "to go together, come together" (see coitus). Sexual meaning "copulation" is attested in English from 1610s (coiture in the same sense is from 1570s).
"coition," early 15c., from Latin coitus "going together," also "coition," from coire "to go together" (see coitus).
late 14c., copulacioun, "a coupling, joining, uniting," from Latin copulationem (nominative copulatio) "a coupling, joining, connecting," noun of action from past-participle stem of copulare "join together, couple, bind, link, unite," from copula "band, tie, link" (see copulate). Specific sense of "sexual intercourse, coition" is from late 15c., and this became the main sense from 16c.
Old English læppa (plural læppan) "skirt or flap of a garment," from Proto-Germanic *lapp- (source also of Old Frisian lappa, Old Saxon lappo, Middle Dutch lappe, Dutch lap, Old High German lappa, German Lappen "rag, shred," Old Norse leppr "patch, rag"), of uncertain origin.
Sense of "lower front part of a shirt or skirt" led to that of "upper legs of seated person" (c. 1300). Used figuratively ("bosom, breast, place where someone or something is held and cherished") from late 14c., as in lap of luxury (which is first recorded 1802). To drop or dump something in someone's lap "shift a burden" is from 1962. From 15c.-17c. the word (often in plural) was a euphemism for "female pudendum," but this is not the source of lap dance, which is first recorded 1993.
To lap dance, you undress, sit your client down, order him to stay still and fully clothed, then hover over him, making a motion that you have perfected by watching Mister Softee ice cream dispensers. [Anthony Lane, review of "Showgirls," New Yorker, Oct. 16, 1995]
Lap-clap was old slang for "an act of coition" (c. 1600), in warning expressions to youth often paired with lip-clip "a kiss." Also compare slang Lapland "the society of women."