"hybrid offspring of donkey and horse," from Old English mul, Old French mul "mule, hinny" (12c., fem. mule), both from Latin mulus (fem. mula) "a mule," from Proto-Italic *musklo-, which is probably (along with Greek myklos "pack-mule," Albanian mushk "mule) a loan-word from a language of Asia Minor.
The mule combines the strength of the horse with the endurance and surefootedness of the ass, and is extensively bred for certain employments for which it is more suited than either; it is ordinarily incapable of procreation. With no good grounds, the mule is a proverbial type of obstinacy. [OED]
Properly, the offspring of a he-ass and a mare; that of a she-ass and a stallion is technically a hinny. The males are ordinarily incapable of procreation. Used allusively of hybrids and things of mixed nature. Meaning "obstinate, stupid, or stubborn person" is from 1470s; the sense of "stupid" seems to have been older, that of "stubborn" is by 18c.
As a type of spinning machine, it is attested from 1793 (as mule-jenny, 1788), so called because it is a "hybrid" of Arkwright's drawing-rollers and Hargreaves' jenny. The underworld slang sense of "narcotics smuggler or courier for a drug trafficker" is attested by 1935. The mule-deer of Western U.S. (1805) is so called for its large ears.
"loose slipper," 1560s, from French mule "slipper," from Latin mulleus calceus "red high-soled shoe," worn by Roman patricians, from mullus "red" (see mullet (n.1)). Related: Mules.