Despite the name, the moon doesn't actually turn blue; and the full moon otherwise isn't much more notable than had the full moon been two weeks ago.
On average the blue moons occur about 4-5 times a decade, and then once every 19 years, you get two in the same year! Last time that happened was in 1999. This July is the first blue moon since August 2012. The next one won't be until January 2018 -- and there will also be one in March.
You can find out when the blue moons are with this handy Blue Moon Calculator by obliquity.com. One caveat -- blue moons can be different months depending on your time zone, if for example it happens at 2 a.m. Eastern Time on Aug. 1 but July 31 at 11 p.m. Pacific Time, the blue moon for Seattle will be in July but in New York, it'll be August. (No issues in the U.S. this year, Seattle's blue moon happened at 4:41 a.m. PDT Friday.)
How we define a blue moon now is actually not how it was originally intended. It was supposed to be the name for a moon when you have four full moons in a three-month period -- the third full moon was then known as the blue moon. An article in Sky & Telescope magazine in 1943 had an error in the original interpretation and gave it the definition we know of today.
The error was discovered by the magazine's editor in 1999, but the cat is way out of the bag now, so the two in a month usage has since trumped the four in a season definition as popular use.