early 15c., "spring of water that collects in a pool," from Old French fontaine "natural spring" (12c.), from Medieval Latin fontana "fountain, a spring" (source of Spanish and Italian fontana), from post-classical noun use of fem. of Latin fontanus "of a spring," from fons (genitive fontis) "spring (of water)," from PIE root *dhen- (1) "to run, flow" (source also of Sanskrit dhanayati, Old Persian danuvatiy "flows, runs").
The extended sense of "artificial jet of water" (and the structures that make them) is first recorded c. 1500. Hence also fountain-pen (by 1823), so called for the reservoir that supplies a continuous flow of ink. "A French fountain-pen is described in 1658 and Miss Burney used one in 1789" [Weekley]. Fountain of youth, and the story of Ponce de Leon's quest for it, seem to have been introduced in American English by Hawthorne's "Dr. Heidegger's Experiment" (January 1837).
"Did you never hear of the 'Fountain of Youth'?" asked Dr. Heidegger, "which Ponce de Leon, the Spanish adventurer, went in search of two or three centuries ago?"