Yes it does but the time for one rotation (one moon day) is the same as the time it takes the moon to orbit the Earth (about 28 Earth days). As a result the moon always has the same face pointing at the Earth.
The moon used to spin faster, but the gravitational 'pull' of the earth has gradually slowed it down, and eventually stopped the moon from spinning. That is why we see only one face of the moon.
Yes, gef is correct it's day is the same length as it takes to do one orbit, what is known as "Tidal Locking" We always see the same side and the other side is often incorrectly called the "dark side" but it gets sunlight.
Tidal lock is a result of friction created within as a changing direction of gravitational pull rearranges interior components. This slows down the rotation until it is in sync. A simular effect is gradually slowing the rotation of the Earth.