During a New Moon the side of the moon that reflects the light from the sun is facing away from the earth. Making the moon barely visible.
Waning Third Quarter
The Waxing First Quarter Moon is positioned so that the side that catches the light from the sun is only half visible. Making the moon look like it is only half of itself.
The moon's gone again?
In a full moon the sun hits the moon in the spot where humans on earth can see all of the parts That are sunlit. Making it bigger than any normal moon in the cycle
Bonus Moon: Waning Crescent
This Moon is nearly the same as the Waxing first quarter moon but the side that is lit up is the opposite of the First Quarter Moon.
See Box 1
As the moon first begins to Wax it becomes a Crescent. The sun hits the moon at an angle so that only a sliver of the lit up part is showing. Fun Fact: A Waning Crescent moon is the one seen in the Dreamworks opening scene