The moon goes around the earth. Sometimes it is on the opposite side as the sun, which gives a full moon. Sometimes it is on the same side as the sun – then you can see it during the day! Most of the time, it is somewhere in between – then you can sometimes see it during the day, and you get a crescent moon at night.
The same reason it is out at night! Light is reflected from the sun on the moon to us… As Willem says, the position of the sun and moon will alter what we see and what shape it is as the amount of light reflected changes!
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