A prude, a puritan, a sanctimonious twit obsessed with other people's morality.
See Also: Donald Wildmon, Tipper Gore, Bill Bennett, Margaret Dumont in the old Marx Brothers movies, the principal who send girls home for wearing skirts above the knee, that old guy who sends letters to the editor complaining about nudity on "The Simpsons."
Where's the term come from? There are several different possibilities. "Bluenose" appears to be a common nickname for people from Nova Scotia, but unless Nova Scotians are notably more puritan than the norm, I doubt it comes from there. Bluenose is also American slang for a Scottish Presbyterian, which holds a bit more promise for being the origin of the word.
However, some say that the term is derived from the "blue laws" that used to hold sway over much of the U.S. In some areas, blue laws regulated whether you were allowed to buy alcohol, buy a car, or go dancing on Sundays -- in other places, blue laws pretty much required all businesses to close down completely on the Lord's Day.
On the other hand, some sources say the blue laws were actually named for the bluenoses who were behind them, not vice versa.
I also suspect that the term could come from the use of "blue" as slang for "pornographic," as in the blue movies that get shown at bachelor parties. And it could come from simple humorous description -- can't you just see some uptight puritans marching primly past the location of some devious sinners? Noses in the air, so terrified of sex or liquor or impropriety that their faces are seemingly drained of passion, blood, and fleshy color? Can't you see the guys at the bar coining the term "bluenoses" as they take another slug of beer?
(I thought of noding this because I was sitting at my desk recently making fun of the company that runs my workplace's intarweb-blocking software. Like most blocking software, there's no rhyme or reason to what it blocks and what it lets get by. Its name is Bluecoat so I started calling it Bluenose. "What's 'bluenose' mean?" asked my coworkers. Hence, the necessity of digging up what little info I could about this word.)
Research:
The dictionary on the bookshelf
http://www.snopes.com/language/colors/bluelaws.asp