Why does the Moon have Phases?
Look at the Moon, and you’ll see that it changes over time. It goes through phases, in a regular cycle. New, crescent, half, gibbous, full, gibbous, half, crescent, and back to new.
It's a common misconception that the phases of the moon is caused by the shadow of the Earth. While the shadow of the Earth does sometimes darken the Moon, that's called an eclipse and it only happens a couple times a year.
So, what does cause it? The short answer is that it's our perspective of the illuminated half of the moon as it moves around us. Perfectly clear, right? Let's break that down.
First, the Moon doesn't generate its own light. It's illuminated by the sun. Here's an experiment you can do at home. Take a ball, go into a windowless room, and turn on a single lamp. Notice that the side of the ball that's facing the lamp looks brighter than the half that's facing away. The side that's facing away isn't completely dark, because there's light bouncing off the walls, but there are no walls in space. The Moon is lit up by the Sun in the same way as the ball is lit up by the lamp.
Half of the Moon is lit up, and half is dark. That's always the case, even during the full moon (except during an eclipse). So, why doesn't it always look like it's half lit up?
Go back to the ball in the room. If you stand between the ball and the lamp (but without getting in the way and casting a shadow) and you look at the ball, what do you see? The illuminated half of the ball. You can't see the dark half, because it's on the other side of the ball.
If you stand so that the ball is between you and the lamp, then you can only see the dark side, because now that's the side that's facing you. And if you stand off to the side, you can see part of the ball that's illuminated, and part that's dark.
Here's a diagram of what I just described. The light source is the yellow circle with rays on the right. The half of the ball facing the light is illuminated. The half facing away is dark. There are three observers. A only sees the light half of the ball. B only sees the dark half. C sees half of each.
Of course, the Earth doesn't revolve around the Moon. the Moon revolves around the Earth, but the principle is the same. When the Moon is between the Earth and the Sun, we see the dark half, so it's a new moon. When the Moon is on the other side of the Earth, we see the illuminated half, so it's a full moon. When the Moon is off to the side, we see some of each.
An implication of this is that you can't see any phase at any time of day. You'll never see a crescent moon at midnight, for example. Because in order for the Moon to appear crescent, it has to be closer to the Sun than the Earth, and at midnight, you're facing directly away from the sun. A full moon rises as the Sun sets, and sets as the Sun rises. A new moon rises and sets at about the same time as the Sun.