Mass is the amount of "stuff", or matter, and object has and this amount remains the same wherever the object is, even though it's weight will change depending upon gravity.
The mass is distributed around the whole object, but if we try and balance the object there is only one point where it will balance perfectly - and this is the Centre of Mass, the place where the mass is concentrated. This point is also called the centre of gravity.
When we have objects orbiting around each other, like the Earth and Moon, the centre of mass is the point around which the objects orbit, and it will always be closer to the more massive object. If the objects are the same mass then the centre of gravity will be exactly half way between the objects.
A good way to picture this is to imagine that we can connect the Moon and Earth with a pole, the centre of mass, or gravity, is the point where we would put our finger to make the pole balance.
As the centre of mass always lies between two objects, this means that both of the objects orbit around it. So even our Sun does not sit still, it orbits around the centre of mass created by all of the planets in our Solar System. All stars with planets going around them do the same, and we often see this as a "wobble", because the centre of mass is so close, or within the star itself. This is one of the main ways we detect exoplanets in our galaxy. For more information on detecting exoplanets click here.