In order for the moon to always show us the same face, it has to rotate one time on its axis for every revolution it makes around us. The effect is caused by tidal forces. The moon's gravity causes tides on earth because the moon's gravity is pulling a little bit harder on the part of earth that faces the moon than on the part of earth that is away from the moon. Since the earth spins, the part of the earth that receives the most pull from the moon's gravity is always changing. For more about tides, visit our friend the Bad Astronomer .
Earth is a little fatter around the equator.
The tidal forces on earth cause friction, as entire oceans shift back and forth. This has caused the earth's rotation to slow down slightly over the years. Our day is a little bit longer than Galileo's was. Synchronous rotation of the moon happens because the moon is not quite round. It is shaped a little like a football or a watermelon.