OED cites a 1914 use and defines it as “Smart, stylish, splendid, luxurious. Also (chiefly Brit.): typical of or belonging to the upper class; (affecting to be) superior or genteel; ‘snooty’, pretentious.” The closest American equivalent would be fancy, or maybe fancy-schmancy.
Posh has been used in the U.S. for decades, of course, but, until the mid-1990s, at roughly half the rate as in Britain and chiefly without the ironic or sardonic connotation noted by the OED. (It’s a similar case to brilliant, in that the word was used here but more sincerely than across the pond.) What happened in the mid-1990s? The Spice Girls, of course. The group was formed in 1994, but it was two years later that Melody Maker magazine bestowed nicknames on the members. Victoria Adams (now Victoria Beckham) actually had an upper-middle-class (what the Brits would call middle-class) upbringing, but she was dubbed Posh Spice because of her bearing, which was, well, posh.
In his article ex-Buster [Thurman] Arnold judicially recorded his opinion that labor has become a national headache, that it is perhaps more unpopular in the slit trenches of World War II than in the posh clubs of professional New Deal haters, and that the great body of public approval essential for effective labor support is crumbling all along the line. (Time, October 18, 1943)/Discerning the difference between “posh” and “body hugger” denims was like trying to tell the Olsen twins apart. (New York Times, May 11, 2011)
notoneoffbritishisms.com 
I doubt this is true, but I’ve heard that posh is actually an acronym. The story goes that it refers to the type of room that one gets on a cruise. As I remember, the story was that “P” stands for portal and “O” either stands for ocean or out(ward). I think it was that the more expensive rooms had portals looking out onto the ocean. My memory is fuzzy on this.
– And that’s the way “facts” are generated on the internet!
Mike: The OED says “unknown” etymology but speculates that it may have derived from an earlier British slang word for “money” or, alternatively, from an Urdu term meaning “well dressed” that is “also used as a colloquial and derogatory term for ‘affluent.’” Then it gets to your story:
“A popular explanation (still frequently repeated) is that the word is the initial letters of the phrase port outward, starboard home, with reference to the more comfortable (because cooler) and more expensive side for accommodation on ships formerly travelling between Britain and India. It is often suggested that the Peninsular & Oriental Steam Navigation Company stamped tickets for such cabins on this route with the letters P.O.S.H., whence the word. However, no evidence has been found for the existence of such tickets.” Ouch.
For what it’s worth, I found a 1940 article from Time about British schoolboy slang where “posh” is identified as having come from the military.
Kate Fox in Watching the English “If you want to ‘talk posh,’ you will have to stop using the term ‘posh,’ for a start: the correct upper-class word is ‘smart’. In upper-middle and upper-class circles, ‘posh’ can only be used ironically, in a jokey tone of voice to show that you know it is a low-class word.” (p 78)
I call it borderline, and I definitely think that it was Posh spice that popularized it over here. I like the term because we don’t really have an exact equivalent, so there’s a space for it… I mean, Canadian culture isn’t based on class really, but words like rich/luxury/classy/exclusive don’t really equate to posh, so I find myself using it a lot. I mean, it still has a totally British ring to it, but it works…
To expand upon the ship voyage to and from India, I have heard it said the theory that POSH stands for Port Out, Starboard Home, referring to which side of the vessel your cabin was located. Port Out meant that on the voyage to India your cabin was away from the direct sun, and vice versa of the turn. That way one didn’t get too hot and sweaty, don’t you know.