[mark on skin resulting from a wound or hurt] late 14c., scarre, "trace left on skin by a healed wound, burn, etc.," from Old French escare "scab" (Modern French escarre), from Late Latin eschara, from Greek eskhara, in medical writing "scab formed after a burn," which is of uncertain origin.
The English sense probably shows influence of another noun scar "crack, cut, incision" (Middle English scarre, skar; attested from late 14c. into 17c.), which is from Old Norse skarð and related to score (n.). Figurative sense attested from 1580s. Old English glossed Latin cicatrix with dolhswað, from dolh "wound" + swað "track, trace."
[rocky cliff] 1670s, "bare and broken rocky face of a cliff or on the side of a mountain;" earlier scarre "naked, detached rock or crag" (14c.; 12c. in place-names), from Old Norse sker "isolated rock or rocky patch at the bottom of the sea," from Proto-Germanic *sker- "to cut" (from PIE root *sker- (1) "to cut") on the notion of "something cut off." The word in the Norse sense is attested in English from 1712.