How many hours of nighttime are in one calendar year?

A year is 8765.8127 hours. Although the length of days and nights vary by latitude and season over the course of a year, the time from sunset to sunrise will very nearly equal 12 hours per day on average, so there will be close to 4382.90635 hours of nighttime.

If one needs a more accurate value, one can use the online tables of darkness provided by the United States Naval Observatory (see reference) and add the values for each day (typically in a spreadsheet, using format [h]:mm:ss). This table allows you to specify all the relevant factors: location and year. As an example, the number of hours of nightime in Boston, MA for 2012 is 4316 hours and 49 minutes.

The other complicating factor in trying to answer this question is with respect to the definition of darkness. The crepuscular hours (twilight hours of either dawn or dusk) are strictly speaking neither daylight nor nightime. Thus, to fully answer this question, one would have to define nighttime.