"oily fat of land animals," c. 1300, from Anglo-French grece, Old French gresse, craisse "grease, fat" (Modern French graisse), from Vulgar Latin *crassia "(melted) animal fat, grease," from Latin crassus "thick, solid, fat" (source also of Spanish grasa, Italian grassa), which is of unknown origin. Grease paint, used by actors, attested from 1880. Grease monkey "mechanic" is from 1918.
mid-14c., "smear, lubricate, or anoint with grease or fat," from grease (n.). Sense of "ply with bribe or protection money" is 1520s, from notion of grease the wheels "make things run smoothly" (mid-15c.). To grease (someone's) palm is from 1580s. Expression greased lightning, representing something that goes very fast, is American English, by 1832.