Well picture the half moon. At half moon, the moon is at right angles from the line connecting the sun and the earth. The light from the sun is hitting the moon, and from our vantage on earth, it looks like a tennis ball lit up from one side with a flashlight. Thus we see what appears to be a half disc. At full moon, it like we have our backs to someone who is holding a flashlight, and we're facing a tennis ball (which is a little higher than our heads so it isn't eclipsed by our shadow) and it is illuminated by the flashlight. We see the whole face of the tennis ball that is illuminated.
So if at full moon, the earth is between the sun and moon, then during the day you're on the part of the earth that is facing away from the moon. As the earth continues to rotate, the moon appears at one horizon, while at the time the sun is disappearing over the other horizon.
Gibbous moons occur as the moon is beyond the earth (from the sun) but not quite 180 degrees. Crescent moons occur when the moon is nearer to the sun than the earth.
This phenomenon is related to the way that we can only see Venus in the morning or evening. Venus's orbit is inside the earth's orbit, so at night when our part of the planet is poitning away from the sun, we're also pointing away from wherever Venus might be. But sometimes in the mornings or evenings, Venus can be seen low in the sky brightly reflecting the light of the sun.
