Here's a paragraph, which I snagged from the Wikipedia entry on "New Weird", which in turn is from the introduction to The New Weird by the editor:
"New Weird is a type of urban, secondary-world fiction that subverts the romanticized ideas about place found in traditional fantasy, largely by choosing realistic, complex real-world models as the jumping off point for creation of settings that may combine elements of both science fiction and fantasy. New Weird has a visceral, in-the-moment quality that often uses elements of surreal or transgressive horror for its tone, style, and effects — in combination with the stimulus of influence from New Wave writers or their proxies (including also such forebears as Mervyn Peake and the French/English Decadents). New Weird fictions are acutely aware of the modern world, even if in disguise, but not always overtly political. As part of this awareness of the modern world, New Weird relies for its visionary power on a "surrender to the weird" that isn't, for example, hermetically sealed in a haunted house on the moors or in a cave in Antarctica. The "surrender" (or "belief") of the writer can take many forms, some of them even involving the use of postmodern techniques that do not undermine the surface reality of the text."
Well. Quite the mouthful. Possibly even a bit pretentious at times. What I get from that, really, is that New Weird eschews the "genre binning" that has separated Horror, Fantasy and Science Fiction, and gleefully combines conventions from all three as needed. I also like the part about being aware of the modern world; some of the earliest High Fantasy (such as The Well at the World's End by William Morris, or even The Lord of the Rings itself) was very much aware of reflecting an earlier period in earth's history; William Morris the "high Middle Ages" and Tolkien the "heroic age" of Germanic sagas. Most later fantasy writers adopted the trappings of the middle ages without adopting the modes of thought and belief, cloaking essentially modern attitudes in a Medieval setting.
New Weird, then, is more self-aware; it doesn't try to be "medieval fantasy"; it's just fantasy and the attitudes of the characters reflect that of the modern author and readers, and in fact do so on purpose.
Although I'm a little skeptical of New Weird as a movement, I'm rather fond of the older concept of the Weird Tale; stories that existed before the calcified divisions that keep science fiction, fantasy and horror as separate genres with separate conventions, and often separate space on the bookshelf of bookstores and libraries. If you take the concept of New Weird to be merely an embrace of that paradigm, without adding pretentious jazz that seem like non-sequiters (why is New Weird called surreal or visceral, for instance? Answer: I dunno, because it isn't. Or at least, isn't necessarily.) then my own tastes are definately leaning towards the New Weird lately. My gaming settings no longer resemble Middle-earth all that much; they resemble Barsoom warmed over with Charles Dickens, H. P. Lovecraft and Sergio Leone. I don't like the "epic quest" plotline very much anymore; I take plot direction from modern thrillers like Robert Ludlum, Ian Fleming or James Patterson.
However, I see a lot of those same things going on in "mainstream" fantasy, not necessarily just stuff that's called New Weird. Scott Lynch's The Lies of Locke Lamora is very Ocean's Eleven-like in terms of plot, for instance. Joe Abercrombie's Best Served Cold is a revenge thriller, part Count of Monte Cristo, part The Magnificent Seven, that just happens to take place in a secondary world, but otherwise isn't too overtly fantastic.
After going through a lengthy adolescence, where fantasy writers were very beholden to trends of the past and the long shadow of writers like Tolkien, Howard, Leiber and maybe Moorcock, finally the genre is growing up, and "mature" books are coming out that take it in new directions.