In one of the internet newsgroups I regularly read, one of the local pedants maintains that a blue moon is the second full moon of a calendar month--a rather rare occurrence. While the descriptivist in me acknowledges that is certainly one of the meanings, the original meaning is more general, referring to any rare occurrence.
The original sense is that of an absurd event that can never occur. The moon is never really blue and once in a blue moon is akin to when pigs fly. (Well actually, when a lot of dust is kicked up into the atmosphere, the moon can appear blue. The eruption of Krakatoa in 1883 caused the moon to turn blue, as did late Indian monsoons in 1927, and Canadian forest fires in 1951.)
The term dates to a 1528 work by William Barlow, the Bishop of Chichester, the Treatyse of the Buryall of the Masse, more commonly known by its first line, Rede me and be nott wrothe, For I say no things but trothe:
Yf they saye the mone is belewe,
We must believe that it is true.
The modern formulation "blue moon" first appears in 1821. Modern usage holds that a blue moon is a rare, but not impossible, event.
The astronomical definition began in 1932 with the Maine Farmer's Almanac. That periodical defined a blue moon as a season with four full moons rather than the usual three. Given that the seasons are defined by the equinoxes and solstices and not the months, this means a year can have twelve full moons, and each month one, yet have one season with four--a blue moon. In 1946, amateur astronomer James Hugh Pruett published an article in which he misinterpreted the Maine rule to mean two full moons in one month. From there the error seems to have propagated--even being repeated in the original game of Trivial Pursuit, which is probably the primary reason for this definition to have spread so widely and quickly.
Two articles on the origins of blue moon appear in Sky and Telescope magazine by folklorist Philip Hiscock and by astronomers Don Olson, Rick Fienberg, and Roger Sinnott respectively in the March and May 1999 issues.