First things first! There are two things we should keep in mind. The first, that the different shapes of the moon are caused by the fact that the moon (or any spherical body) is always half lit by the sun, that is, the side facing the sun is bright and the one facing away is dark. Secondly, the plane of moon’s orbit about Earth is inclined to the Earth’s orbit around the sun by 5.1 degrees.
The lunar cycle starts with the new moon, the new moon is in between the earth and the sun. Yes, it can only be detected during the daytime, but we can’t see it because the dark side of the moon is facing us.
At this point we can stop and define a few words.
Waxing means growing and waning means shrinking. Gibbous means swollen or convex. Terminator is the parting line of day and night in the moon. With that lets continue!
The moon at the new moon phase moves towards the east to get the waxing crescent moon. The crescent line comes because the terminator is curved and we view it at an angle such that we see only a part of the bright side of the moon.
As it finishes quarter of its revolution, we can clearly see half the moon is bright and other half is dark. The terminator is seen straight. There is a misconception here. Although it’s seen as half lit and half dark, it is called quarter moon because it finished quarter of its revolution
After a couple of days, the bright side of the moon grows while the night side fades away. At this point it is called waxing gibbous due to the large swollen bright side.
When the Earth is positioned at the center with the sun and the moon on either side, we can see the full moon during the night.
From now on, the day side of the moon shrinks and the nightside grows. The process is exactly reversed for the rest half of the lunar cycle. The full moon is followed by the waning gibbous where the bright day is in reversed direction. It is then followed by the last quarter; which is half bright. Lastly before the next new moon we observe the waning crescent moon and then the cycle continues like since time immemorial.
We should also realize that since the time period of revolution of moon around earth and moon’s rotation is same the same side of the moon will ALWAYS face us. The night sides of the moon at its different phases are still the same side we see during full moon