Image credit:Kostian Iftica, ExploreTheCosmos.com
But even this name is not technically correct. First, astronomers use Universal Time (Greenwich Mean Time, or the time in Greenwich, England when it is not Daylight Savings Time). And, by that measure, the full moon occurs early in the morning on June 1 (1:04am UT, to be precise). So the astronomical blue moon will be next month, when the second full moon will be June 30 at 1:34pm UT.
Second, using the term "Blue Moon" to refer to the second full moon in a month started as a mistake. This article from Sky & Telescope magazine explains how that mistake was made. In the past, the term "blue moon" referred to the third full moon in a season that had four full moons instead of the normal three. But the mistake is now ingrained in popular culture, and there is no need to try and correct it (in my humble opinion).
Scientifically, there is nothing interesting about a blue moon. It takes the moon 29.5 days to go through a set of phases, and months (except for February) are longer than this. So, it is possible, with the exception of February, to have two full moons in a month. This happens every two and a half years or so, including this (or next) month.
In this way, a blue moon is like Leap Day -- it is an occasional event because our calendar (based on the Earth going exactly once around the sun) doesn't match up exactly with cycles of the moon's phases (a "moonth," if you like) or the Earth's spinning about its axis (a "day").
So, although there is nothing special about this full moon, I am happy for the press. Anything that can get people to go and look at the sky is a bonus. If you go out looking for the blue moon this evening, look for the planet Venus (by far the brightest "star" in the western sky after sunset right now.) The Big Dipper is high in the sky in the early evening, too. Just above and to the right of the moon is the star Antares, the heart of Scorpio. To the left of the moon, the bright "star" is the planet Jupiter. Compare the colors of Jupiter to Antares, and you might be surprised to notice that Antares is definitely reddish in color. (It may be a little hard because of the glare of the moon). And Venus is definitely whiter than either Jupiter or Antares!