c. 1300, presse, "crowd, throng, company; crowding and jostling of a throng; a massing together," from Old French presse (n.) "throng, crush, crowd; wine or cheese press" (11c.), from Latin pressare (see press (v.1)). Late Old English had press "clothes press."
Meaning "device for pressing cloth" is from late 14c., as is also the sense "device to squeeze juice from grapes, oil from olives, cider from apples, etc.," from Middle French presse. Specific sense "machine for printing" is from 1530s; this was extended to publishing houses by 1570s and to publishing generally (in phrases such as freedom of the press) from c. 1680. This gradually shifted c. 1800-1820 to "periodical publishing, journalism." The press, meaning "journalists collectively" is attested from 1921 (though superseded by media since the rise of television, etc.).
Press agent is from 1873; press conference is attested from 1931, though the thing itself dates to at least World War I. Press secretary is recorded from 1940. Via the sense "crowd, throng," Middle English in press meant "in public," a coincidental parallel to the modern phrase in the press. Weightlifting sense is from 1908. The basketball defense so called from 1959 (in full-court press).