fat (adj.) Look up fat at Dictionary.com
Old English fætt "fat, fatted, plump, obese," originally a contracted past participle of fættian "to cram, stuff," from Proto-Germanic *faitaz "fat" (cognates: Old Frisian fatt, Old Norse feitr, Dutch vet, German feist), from PIE *poid- "to abound in water, milk, fat, etc." (source also of Greek piduein "to gush forth"), from root *peie- "to be fat, swell" (cognates: Sanskrit payate "swells, exuberates," pituh "juice, sap, resin;" Lithuanian pienas "milk;" Greek pion "fat, wealthy;" Latin pinguis "fat").

Teen slang meaning "attractive, up to date" (also later phat) is attested from 1951. Fat cat "privileged and rich person" is from 1928; fat chance "no chance at all" attested from 1905, perhaps ironic (the expression is found earlier in the sense "good opportunity"). Fathead is from 1842; fat-witted is from 1590s; fatso is first recorded 1943. Expression the fat is in the fire originally meant "the plan has failed" (1560s).
fat (n.) Look up fat at Dictionary.com
mid-14c.; see fat (v.). Figurative sense of "best or most rewarding part" is from 1560s.