There seem to be a large number of people who didn't get the main point of your question. Only one answer has actually mentioned the common centre of mass, which is correct. This common centre of mass, called the barycentre, is located within the Earth, because the Earth has about 80 times the mass of the Moon. What this means is that the orbit of the Moon around the Earth causes a small wobble in the Earth's motion. Pluto and Charon are so similar in mass that the barycentre lies in space between the two objects, resulting in Pluto doing a strange spiral dance as it orbits the sun.
The reason we see only one side of the Moon, however, is not related to this. It's a phenomenon called tidal lock. This might get complicated, but I'll try to keep it simple. Essentially the Earth's gravity distorts the Moon, creating a bulge. In the past, when the Moon was rotating faster, the spin of the Moon would carry the bulge away from the line between Earth and Moon. Earth's gravity pulls on the bulge, slowing the Moon's rotation. Eventually the Earth's gravity pulls the bulge sufficiently to keep the same side of the Moon facing Earth. keeping the bulge pointed directly to the Earth's centre of gravity. Note that the Moon is still rotating on its axis, just once every orbit.