About the moon's phases
Traditionally, there are eight phases of the moon. They are: The New Moon, Waxing Crescent (also called a "Evening Crescent"), First Quarter, Waxing Gibbous, Full Moon, Waning Gibbous, Third Quarter (or "Last Quarter"), Waning Crescent (also called an "Morning Crescent") and back to the New Moon where the lunar cycle begins again.
There's no such thing as a "half moon"
You've probably heard the term "half moon" many times. The truth is, this isn't actually one of the names for the moon's phases. The First Quarter and Last Quarter moons are sometimes descriptively called "Half Moons", but the correct term is "Quarter Moon" because they represent the points at 25% and 75% of the full lunar cycle.
Tides and the phases of the moon
The Earth's water is drawn towards the moon at all times. But during some phases of the moon, the effect is more pronounced.
During a Full Moon and a New Moon the Earth's tides are highest (often called a "Spring Tide", although the effect happens at any time of year). During each of the moon's Quarter phases, or "Neap Tides" the pull of the moon is the subtlest and Earth's tides are at their lowest.