"principal officer of a municipality, chief magistrate of a city or borough," c. 1300, mair, meir (mid-13c. as a surname), from Old French maire "head of a city or town government" (13c.), originally "greater, superior" (adj.), from Latin maior, major, comparative of magnus "great, large, big" (of size), "abundant" (of quantity), "great, considerable" (of value), "strong, powerful" (of force); of persons, "elder, aged," also, figuratively, "great, mighty, grand, important," from PIE *mag-no-, from root *meg- "great."
Mayoress is attested from late 15c. as "the wife of a mayor;" by 1863 as "woman holding the office of mayor."
word-forming element meaning "quality, condition; act, power, skill; office, position; relation between," Middle English -schipe, from Old English -sciepe, Anglian -scip "state, condition of being," from Proto-Germanic *-skepi- (cognates: Old Norse -skapr, Danish -skab, Old Frisian -skip, Dutch -schap, German -schaft), from *skap- "to create, ordain, appoint," from PIE root *(s)kep-, forming words meaning "to cut, scrape, hack" (see shape (v.)). It often forms abstracts to go with corresponding concretes (friend/friendship, etc.).