Does the moon rotate on an axis like the earth?

Yes, this is why we only see one side of the moon - it rotates at the same rate as it orbits.

The moon rotates around it's own axis while it orbits planet earth.
The moon's rotation is "in synch" with the orbit around earth (1 rotation during 1 orbit).
That is why we see only one side of the moon from earth.

It is interesting to note that although the moon orbits planet earth , planet earth also wobbles around their common path (called "barycenter") around the sun.

Link to barycenter animation earth-moon path around the sun.