1580s, "open public square in an Italian town," from Italian piazza, from Latin platea "courtyard, broad street," from Greek plateia (hodos) "broad (street)," from platys "broad, flat" (from PIE root *plat- "to spread"). According to OED, mistakenly applied in English 1640s to the colonnade of Covent Garden, designed by Inigo Jones, rather than to the marketplace itself; hence "the veranda of a house" (1724, chiefly American English).
c. 1200, "space, dimensional extent, room, area," from Old French place "place, spot" (12c.) and directly from Medieval Latin placea "place, spot," from Latin platea "courtyard, open space; broad way, avenue," from Greek plateia (hodos) "broad (way)," fem. of platys "broad," from PIE root *plat- "to spread."
Replaced Old English stow and stede. From mid-13c. as "particular part of space, extent, definite location, spot, site;" from early 14c. as "position or place occupied by custom, etc.; precedence, priority in rank or dignity; social status, position on some social scale;" from late 14c. as "inhabited place, town, country," also "place on the surface of something, portion of something, part." Meaning "a situation, appointment, or employment" is by 1550s. Meaning "group of houses in a town" is from 1580s.
Also from the same Latin source are Italian piazza, Catalan plassa, Spanish plaza, Middle Dutch plaetse, Dutch plaats, German Platz, Danish plads, Norwegian plass. The word appears via the Bible in Old English (Old Northumbrian plaece, plaetse "an open place in a city"), but the modern word is a reborrowing.
Sense of "a mansion with its adjoining grounds" is from mid-14c.; that of "building or part of a building set apart for some purpose is by late 15c. (in place of worship). Meaning "a broad way, square, or open space in a city or town," often having some particular use or character (Park Place, Waverly Place,Rillington Place) is by 1690s, from a sense in French. Its wide application in English covers meanings that in French require three words: place, lieu, and endroit. Cognate Italian piazza and Spanish plaza retain more of the etymological sense.
To take place "happen, come to pass, be accomplished" (mid-15c., earlier have place, late 14c.), translates French avoir lieu. To know (one's) place "know how to behave in a manner befitting one's rank, situation, etc." is from c. 1600, from the "social status" sense; hence the figurative expression put (someone) in his or her place (1855). In in the first place, etc., it has the sense of "point or degree in order of proceeding" (1630s). Out of place "not properly adjusted or placed in relation to other things" is by 1520s. All over the place "in disorder" is attested from 1923.