Old English bodig "trunk of a man or beast, physical structure of a human or animal; material frame, material existence of a human; main or principal part of anything," related to Old High German botah, but otherwise of unknown origin. Not elsewhere in Germanic, and the word has died out in German (replaced by Leib, originally "life," and Körper, from Latin), "but in English body remains as a great and important word" [OED].
Extension to "a person, a human being" is from c. 1300. Meaning "main part" of anything was in late Old English, hence its use in reference to vehicles (1520s). From 1580s as "part of the dress which covers the body." From 1590s as "main part of a group, any number of individuals spoken of collectively." From 1660s as "main portion of a document." Contrasted with soul at least since mid-13c. Meaning "corpse" ("dead body") is from c. 1200. Transferred to matter generally in Middle English (as in heavenly body, late 14c.).
Body politic "the nation, the state, whole body of people living under an organized government" first recorded late 15c., with French word order. Body image was coined 1934. Body count "number of enemy killed in battle or otherwise" is from 1968, from the Vietnam War. Body language is attested from 1967, perhaps from French langage corporel (1966). Body-snatcher "one who secretly disinters the bodies of the recently dead for dissection" is from 1834. Phrase over my dead body attested by 1833.