c. 1300, "limbless reptile," also the tempter in Genesis iii.1-5, from Old French serpent, sarpent "snake, serpent" (12c.), from Latin serpentem (nominative serpens) "snake; creeping thing," also the name of a constellation, from present participle of serpere "to creep."
This is reconstructed to be from PIE *serp- "to crawl, creep" (source also of Sanskrit sarpati "creeps," sarpah "serpent;" Greek herpein "to creep," herpeton "serpent;" Albanian garper "serpent").
Serpent and snake now mean precisely the same thing ; but the word serpent is somewhat more formal or technical than snake, so that it seldom applies to the limbless lizards, many of which are popularly mistaken for and called snakes, and snake had originally a specific meaning. [Century Dictionary, 1902]
Used figuratively of things spiral or regularly sinuous, such as a type of musical instrument with a twisting tube (1730). Serpent's tongue as figurative of venomous or stinging speech is from mistaken medieval notion that the serpent's tongue was its "sting." Serpent's tongue also was a name given to fossil shark's teeth (c. 1600). Serpent-charmer is by 1861.