Yes, it does. It rotates on its axis in what is called synchronous rotation. This means that it takes the Moon as long to rotate fully on its own axis as it takes it to orbit fully around the Earth (which is about 27-28 days, or what is sometimes called a lunar month). This synchronous rotation is caused by tidal locking - gravitational pull means that the same side of the Moon is always facing the Earth. When people talk about the dark or the far side of the Moon they mean the side that we can't see from Earth.
It is possible to get an idea of the Moon's movement if you watch on a clear night, when over several hours you will see its position change very slightly relative to the stars.
You can learn more about the Moon's orbit and rotation here.