The first moon is the new moon. At new moon, the Moon is lined up between the Earth and the Sun. We see the side of the Moon that is not being lit by the Sun. When the Moon is exactly lined up with the Sun we experience an eclipse.
The second moon phase is the waxing cresent. At new moon, the Moon is lined up between the Earth and the Sun. We see the side of the Moon that is not being lit by the Sun. When the Moon is exactly lined up with the Sun, we experience an eclipse.
The third moon phase is the first quarter. When the moon is at its first quarter, the right half is lit up. This phase is between the waxing crescent and waxing gibbous. It is called the first quarter because it is a quarter of the way through the moon phase cycle.
During the last quarter, the left half is lit up.
Sometimes this is also called the third quarter. The last quarter moon phase is between the waning crescent and the waning gibbous.
a waning crescent, we can only see a small piece on the left that is lit up. After this phase is complete, the moon's cycle will start over again with the new moon.