Why does the Moon change shape?
We see different amounts of its sunlit half as it circles the Earth.
Half of the Moon is always lit by the Sun. When the Moon is between us and the Sun we see the dark side: new Moon. When it's on the far side of Earth we see the whole lit face: full Moon.
In between we see a slice: crescent, quarter, gibbous. The whole cycle takes 29.5 days. It's not Earth's shadow — that only happens during a lunar eclipse.
Related questions
Why does the Moon rise later every day?
While the Earth spins, the Moon also moves along its orbit, so the Earth needs about 50 extra minutes to catch up with it.
What is an eclipse?
When the Sun, Earth and Moon line up exactly, one casts a shadow on the other.
Why can I see the dark part of a crescent Moon?
Sunlight reflected off Earth lights up the Moon's night side — earthshine.