Two New Moons in a Month
The most common use of the term "black moon" relates to the frequency of new moons. A black moon (in this context) refers to the second new moon to occur in a calendar month. A black moon cannot occur in the month of February.
Third New Moon in a Season
"Black moon" may also describe the third new moon in a season of four new moons. This definition is taken from the Maine Farmer's Almanac. The names for the other three moons (of four) in a season are the secret moon, finder's moon and spinner moon.
A Month With No Full Moon
The absence of a full moon in a calendar month is also known as a "black moon." The only time a black moon can occur in this context is in the month of February. Additionally, with the occurrence of a black moon, either January or March must have two full moons.
A Month With No New Moon
The fourth use of "black moon" is to describe a calendar month without a new moon. Similar to a month without a full moon, February is the only month in which there cannot be a new moon. In the event of no new moon in February, either March or January must have a second new moon.
Rarity of a Black Moon
In all of the four contexts, the occurrence of a black moon is quite rare. As with a calendrical blue moon, black moons occur once every 2.5 years in the form of two new moons in a month. The most rare black moon--that which is a February without either a full or a new moon--happens once about every 19 years; the last occurred in 1999, and the next will be in 2018.