A phenomenon like blue moon is a full moon that can be seen twice per month. It is not really blue, but there is a possibility that it can become a little blue. The definition of the term blue moon as we use it today was created due to the writer's mistake in the past.
The phrase "blue moon" itself is a fruit of folklore imagination, according to Phillip Hiscock (folklorist at Memorial University of Newfoundland). He says that it is a second full moon in a row during one month.
Phillip helped us know the origin of this term. In past, the phrase "blue moon" meant something unreal and foolish and unbelievable. However, in 1946 a non-professional astronomer James H. Pruett was publishing in "Sky and Telescope" magazine. He repeated the opinion of Maine Farmers' Almanac that blue moon is third full moon in a season (meaning season to have 4 of them, instead of right 3). The error has been later copied and transmitted through a radio program in 1980.
Hiscock with an astronomer Donald W. Olson revealed the mistake and helped to correct it in 1999. However, the mistake became widely adopted by people, and nowadays it is associated with a second full moon in a month. This phenomenon appears quite rarely - once in 2.7 - 3 years.
Famous collocation "once in a blue moon" which means something really rare, originated in the beginning of 19th century ( ≈ 1924).
However, is it true that the Moon can be blue? Researchers say it is possible.
In case of essential volcanic eruption or massive forest fires when large masses of smoke, ash uplift in the near space, the Moon can really get some blue undertones. An example of something like this happened in 1950, when Canadian forest was burning and a blue moon could have been seen over North America. The great eruption of Pinatubo Mount in Philippines was so huge that caused blue moon that could have been seen almost all over the world.