"thing to sit on; act of sitting," c. 1200, from Old Norse sæti "seat, position," from Proto-Germanic *sæt- (source also of Old High German saze, Middle Dutch gesaete "seat," Old High German gisazi, German Gesäß "buttocks"), from PIE root *sed- (1) "to sit." Meaning "posterior of the body" (the sitting part) is from c. 1600; sense of "part of a garment which covers the buttocks" is from 1835. Seat belt is from 1915, originally in airplanes.
"residence, abode, established place," late 13c., extended use of seat (n.1), influenced by Old French siege "seat, established place," and Latin sedes "seat." Meaning "city in which a government sits" is attested from c. 1400. Sense of "right of taking a place in a parliament or other legislative body" is attested from 1774. Old English had sæt "place where one sits in ambush," which also meant "residents, inhabitants," and is the source of the -set in Dorset and Somerset.