The phases of the moon are caused by the relative positions of the sun, Earth and moon. The moon travels around the Earth, taking just over 29 days to complete a single orbit. The sun illuminates the moon from various angles as it changes position relative to the Earth.
The light of the moon is actually sunlight that is reflected off the lunar surface. While sunlight is always falling on half of the moon, the potion of the moon's surface that appears illuminated from the Earth changes depending on where the moon is located in its orbit. The moon goes through phases from the new moon to full moon and then back again.
The moon is said to be waxing as more and more of its surface becomes illuminated. The crescent-moon phase occurs when less than half of the moon is illuminated. First-quarter moons are when the moon is at a 90-degree angle from the Earth and the sun, at which point half of the moon's surface is illuminated by sunlight while the other half remains in shadow, as seen from the Earth. The gibbous phase occurs before the moon's disk is fully illuminated. After the full moon, the moon wanes back through the cycle.