c. 1300, presse, "a crowd, throng, company; crowding and jostling of a throng; a massing together," from Old French presse (n.) "a throng, a crush, a crowd; wine or cheese press" (11c.), from Latin pressare (see press (v.1)). Late Old English had press in the sense of "clothes press," but the Middle English word probably is from French.
The general sense of "instrument or machine by which anything is subjected to pressure" is from late 14c.: "device for pressing cloth," also "device to squeeze juice from grapes, oil from olives, cider from apples, etc." The sense of "urgency, urgent demands of affairs" is from 1640s. Weightlifting sense is from 1908. The basketball defense so called from 1959 (in full-court press).
The specific sense "machine for printing" is from 1530s; this was extended to publishing houses and agencies of producing printed matter collectively by 1570s and to publishing generally (in phrases such as freedom of the press) from c. 1680. This gradually shifted c. 1800-1820 to "the sum total of periodical publishing, newspapers, journalism." The press, meaning "journalists collectively" is attested from 1921 (though superseded by media since the rise of television, etc.).
Press agent, employed to tend to newspaper advertisements and supply news editors with information, is from 1873, originally theatrical; press conference "meeting at which journalists are given the opportunity to question a politician, celebrity, etc.," is attested from 1931, though the thing itself dates to at least World War I. Press secretary is recorded from 1940; press release "official statement offered to a newspaper for publication" is by 1918.
Via the sense "crowd, throng," Middle English in press meant "in public," a coincidental parallel to the modern phrase in the press.