"female of the horse or any other equine animal," Old English meare, also mere (Mercian), myre (West Saxon), fem. of mearh "horse," from Proto-Germanic *marhijo- "female horse" (source also of Old Saxon meriha, Old Norse merr, Old Frisian merrie, Dutch merrie, Old High German meriha, German Mähre "mare"), said to be of Gaulish origin (compare Irish and Gaelic marc, Welsh march, Breton marh "horse").
The fem. form is not recorded in Gothic, and there are no known cognates beyond Germanic and Celtic, so perhaps it is a word from a substrate language. The masc. forms have disappeared in English and German except as disguised in marshal (n.). In 14c. also "a bad woman, a slut," and, apparently, also "a rabbit." As the name of a throw in wrestling, it is attested from c. 1600. Mare's nest "illusory discovery, something of apparent importance causing excitement but which turns out to be a delusion or a hoax" is from 1610s.
"broad, dark area of the moon," 1765, from Latin mare "sea" (from PIE root *mori- "body of water"). Applied to lunar features by Galileo and used thus in 17c. works written in Modern Latin. They originally were thought to be actual seas.
"night-goblin, incubus, oppressed sleep," Old English mare "incubus, nightmare, monster," from mera, mære, from Proto-Germanic *maron "goblin" (source also of Middle Low German mar, Middle Dutch mare, Old High German mara, German Mahr "incubus," Old Norse mara "nightmare, incubus"). This is from PIE *mora- "incubus" (source also of first element in Old Irish Morrigain "demoness of the corpses," literally "queen of the nightmare," also Bulgarian, Serbian mora, Czech mura, Polish zmora "incubus"). Also compare French cauchemar "nightmare," with first element from Old French caucher "to trample" and second element from Germanic.
All this is probably from PIE root *mer- "to rub away, harm" (also "to die" and forming words referring to death and to beings subject to death). The word in English now survives only in nightmare (q.v.).