A quarter moon occurs when we see the Moon half illuminated by the Sun, and half enshrouded in darkness. Since the illuminated side points towards the Sun, it tells astronomers that the Moon and the Sun are separated by 90-degrees from our perspective here on Earth.
You can have a first quarter moon, when the Moon is halfway between a new moon and full moon. A new moon occurs when the Moon is directly in between the Sun and the Earth. From that perspective, the illuminated Moon is facing away from the Earth. A full moon happens when the Moon and the Sun are on opposite sides of the Earth. At that point, we see the Moon fully illuminated by light from the Sun.
When we see the first quarter of the Moon, that means the amount of the Moon that’s illuminated is increasing. Astronomers call an increasingly illuminated moon: “waxing”.
You can also get a last quarter moon, when the Moon is half illuminated, but the opposite side is illuminated. This is the halfway point between a full moon and a new moon. When the amount of the Moon that’s illuminated is decreasing, astronomers call this “waning”.
We have an article that explains all the phases of the Moon.
Want to know when the next quarter moons are going to happen? Here’s a chart from NASA that calculates the phases of the Moon over a 6000 year period. And here’s a cool calculator that shows you what the Moon is doing right now.
You can listen to a very interesting podcast about the formation of the Moon from Astronomy Cast, Episode 17: Where Did the Moon Come From?