Why is the moon not visible sometimes?

There are several reasons the moon may not be visible:

1) It is over the horizon from the person observing it - past their line of sight

2) The view of the moon is blocked by something such as clouds (or walls if you are inside)

3) During a "new moon" the side of the moon that is illuminated is opposite the side that would be observed from the surface of the earth - and it is rising and setting with the sun. The moon may still receive a small amount of light reflected from the earth, but under these conditions, the moon is not visible because without that illumination from the sun, the sun is so much brighter than the feeble re-reflected light (from the sun to the earth to the moon) that the moon is not visible. A dramatic example of this is when the moon lines up with the sun to produce a solar eclipse.