"cylinder of wood or metal with a spiral ridge (the thread) round it," c. 1400, scrue, from Old Frenchescröe, escroue "nut, cylindrical socket, screw-hole," a word of uncertain etymology; not found in other Romanic languages.
Perhaps via Gallo-Roman *scroba or West Germanic *scruva from Vulgar Latin scrobis "screw-head groove," in classical Latin "ditch, trench," also "vagina" (Diez, though OED finds this "phonologically impossible"). OED Seems to lean toward a group of apparently cognate Germanic words (Middle Low German, Middle Dutch schruve, Dutch schroef, German Schraube, Swedish skrufva "screw"), but these are said elsewhere to be French loan-words.
Kluge, Watkins and others trace it to Latin scrofa "breeding sow," perhaps on some fancied resemblance of the holes or furrows left by a rooting swine (compare Portuguese porca, Spanish perca "a female screw," from Latin porca "sow"). Latin scrofa in the "sow" sense is a specific Medieval Latin use; the word is literally "digger, rooter" (from PIE root *sker- (1) "to cut").
Originally an apparatus for lifting weight or pressing with it, hence its later consideration as one of the mechanical powers. The meaning "metal pin or tapered bolt with a spiral ridge, used to join articles of wood or metal," is by 1620s (specifically as wood-screw by 1841). The meaning "a twist or turn to one side" is by 1709.
The sense of "means of pressure or coercion" is from 1640s, often the screws, probably in reference to instruments of torture (as in thumbscrews). Meaning "prison guard, warder" is by 1812 in underworld slang, originally in reference to the key they carried (screw as slang for "key" is attested by 1795). In student slang, "professor or tutor who requires students to work hard" (1851).
The meaning "metal instrument with a winding or spiral shape or motion, used to draw corks from bottles" is by 1650s. As short for screw-propeller, by 1838. The sense of "small portion (of a commodity) wrapped up in a twist of paper" is by 1836. The British slang sense of "salary, wages: is by 1858, but the notion in it is obscure. The slang meaning "an act of copulation" is recorded from 1929 (canting sense of "a prostitute" is attested from 1725). Slang phrase have a screw loose "have a dangerous (usually mental) weakness" is recorded from 1810.