You have spoken. Earlier this week, I asked for nominees for the next new NOOB, and you chose (narrowly) twee. So twee ’tis.
As for definition, I can do no better than the most popular entry at urbandictionary.com:
Something that is sweet, almost to the point of being sickeningly so. As a derogatory descriptive, it means something that is affectedly dainty or quaint, or is way too sentimental. In American English it often refers to a type of simple sweet pop music, but in British English it is used much more widely for things that are nauseatingly cute or precious. It comes from the way the word sweet sounds when said in baby talk.“Belle and Sebastian are the Beatles of twee.”
The OED gives the etymology of twee as “infantile” pronunciation of sweet, dates it to 1905, and offers this definition: “Originally: ‘sweet’, dainty, chic. Now only in depreciatory use: affectedly dainty or quaint; over-nice, over-refined, precious, mawkish.”
Americans may find [choreographer Ronald] Hynd all too twee (look that one up in your Anglo-American dictionary). (Anna Kisselgoff, New York Times, April 18, 1982)/[Mari Eastman’s] paintings are pretty and a bit twee, and I never found them very interesting, until now. Culture Monster [Los Angeles Times blog], April 21. 2011)
notoneoffbritishisms.com 
Google Trends Help claims the y-axis and other features are visible if you’re registered and logged in, and that search traffic is scaled to 1.0. Their news reference volume supposedly shows the number of times a topic appeared in Google News stories, but is only “illustrative”–moreso, I found, if I put in two terms: http://www.google.com/trends?q=twee%2C+cute&ctab=0&geo=all&date=all&sort=0
It seems that the use of twee is found mostly in reviews. it has become more frequent in movie reviews over the last few years I have noticed and in many reviews of “emo” bands (who’d’ve thunk?)
Most English people wouldn’t be caught dead using the word “twee” in everyday speech. It’s a very old fashioned term.
It’s rather twee here, too.
To me, “twee” means specifically something that is old fashioned and sweet/cute. So somebody who is twee is somebody who appears old fashioned.