Explain why we see different portions of the Moon illuminated in its cycle, from our vantage point on Earth.

Thank you.

The moon circles the Earth... at times, it's closer to the sun than we are, at other times, we're closer to the sun.

Starting at New Moon, the moon is closest to the sun in it's orbit about Earth. That means that the side of the moon illuminated by the sun is on side facing away from Earth, so only the shadowed side is facing us.
As the moon continues in it's orbit, we begin to see the side being lit by the sun, and from our viewpoint, the moon will be crescent-shaped.
When the moon and Earth are the same distance from the sun, we see 1/2 the disk of the moon being lit by the sun's light, so we see the half-moon. (called the 1st quarter.
It keeps moving around us, and the amount of the moon's surface we see being lit increases, giving us a 'gibbous' moon - more than half, but not quite full.
When the moon is furthest from the sun, we see the full lit face of the moon - giving is the 'full moon.'
Now, as it moves back towards it's starting point, it goes through the 'gibbous' phase again, as the shadow begins to creep back across it's face. When the moon is back to being the same distance to the sun as Earth is, we see the 1/2 moon again, only - it's the *other* half being lit.
After half-moon, it turns back into a crescent moon, which gets thinner and thinner until we're back to full moon.

I think the links to this video and this picture is really going to help explain it better than someone writing an answer:


http://www.youtube.com/watch?v=YdI1aDjWL…
http://www.moonconnection.com/images/moo…