During the new moon phase, the moon is positioned between the Earth and the Sun. It actually appears to people on the Earth that there is no moon at all during this phase. The reason for this optical illusion is that the portion of the moon lit by the sun is facing away from Earth.
When the moon is full, the Earth, moon, and sun are in approximate alignment as with the new moon. During this alignment, however, the Earth is between the moon and the sun, and the moon is positioned on the opposite side of the Earth. This positioning allows people in California to see the entire face of the moon as it is lit by the sun, while the unlit portion is facing away from the Earth.
The first and third quarter moon phases are more commonly referred to as a "Half Moon." This is because the position of the moon in relation to the Earth and the sun is a 90-degree angle. This angle only allows for half of the moon's face to be lit at one time.
After the new moon, the light on the moon is increasing, and this phase is called the waxing crescent. It is called crescent because less than half of the moon's face is lit, thereby giving the impression of a crescent-shaped moon. After the first quarter, the light increases on the face of the moon, but more than half of the face is lit during this phase. This almost full appearance of the moon is called waxing gibbous.
As the new moon phase approaches and the third quarter phase ends, the moon is a waning crescent. The classic crescent moon shape is present, and the light is decreasing on the face of the moon until it disappears, totally ushering in the new moon phase. The waning gibbous phase occurs after the full moon as the light on the face of the moon begins to decrease. The waning gibbous phase gives the appearance of a near full moon, because more than half the face of the moon is bathed in sunlight.