We see the Moon because it reflects the Sun's light. We only see the part of the Moon that is lit up by the Sun. The other part remains dark. The Moon receives the Sun's light from different angles as it orbits the Earth. This makes it look like the Moon is changing shape. We call these changes the phases of the Moon.
It takes 29.5 days to go from one full moon to the next. Have you noticed this is slightly longer than the time it takes to orbit the Earth? This is because the Earth is moving around the Sun. The Moon has to travel a bit further to get back to the same position.
The Moon is "tidally locked" into position with the Earth. One side always faces the Earth, and the other side always faces away. This is because it takes the same amount of time to spin on its axis once as it does to orbit the Earth once. The gravity between the Earth and the Moon caused this special case of tidal locking (called synchronous rotation).