When talking about the Wheel of the Year the focus is usually on the solar cycle, but the moon cycles of the year also can have names, and celebrate specific seasonal events as well.
There are a few different ways to name the moon cycles. First is by the month, which is the way more people are probably familiar with. With this method, whatever month the full moon is in determines the name of the cycle (which starts at the new moon).
To give an example, say the name of April's moon is the Pink Moon. There is one full moon in April, and it lands on the 28th. That would mean the cycle that began with the new moon on April 14th is the Pink Moon cycle - even though it would end in May. Simple enough, right? So what happens when there are two full moons in a month? The second cycle would be Blue Moon. (This practice actually stems from a misunderstanding, read on for the original meaning!)
Some people prefer to modify this just a bit, and begin the cycle with where the new moon is in the month. This wouldn't change the example above, but you would find a difference if April's full moon was on the 5th. Using the full moon cycle this would still be the Pink Moon, but using the new moon cycle it would take on March's name instead.
Another method uses the seasons. First you identify when the solstices and equinoxes are, then when the moons fall around them. Each season will generally have three full moons. Looking at the summer solstice, say it lands on June 21st, and there is a full moon three days later - that is the first of the three summer moons. If the first full moon isn't until July 2nd, it is still given the name of the first summer moon, regardless of the month it ends up in. Sometimes, though, a season will have four full moons. The third moon of a cycle of four will be the Blue Moon - this is the original definition of a Blue Moon.
So where did the monthly definition come from? Interestingly, it came from a mistake made in an issue in Sky & Telescope magazine printed in 1946. The writer of the article misunderstood his source, the Maine Farmer's Almanac. (This particular Almanac also used fixed dates for the solstices and equinoxes.) Still, there's no doubt that the monthly definition is more commonly known these days. Both systems have their pros and cons, and it's worth it to explore both.
So, we've talked a bit about the Blue Moon, but what about the other moon names? Let me be very clear - there are many different names for each moon cycle, coming from many different locations. The moon names usually reflect something going on during that time of year, which of course would change from location to location. As an example, the April moon mentioned earlier? Pink Moon is just one name, it can also be the Seed Moon, Sprouting Grass Moon, Egg Moon, or the Fish Moon, among others. Due to this, you'll sometimes see slight differences in which month is which name. Seed moon can be April's Moon, but it can also be March's. There are many different calenders which can be explored, and it can be helpful to find or make one that fits your local cycles.