Center of mass or barycenter of a system is the average position of all the mass in a system.
In a rigid body, the centre of mass is always in the same place. In a loose collection the center of mass may be in space, as it is in the Solar System. Also, as planets' orbits change slightly, so will the position of the barycenter.
A simple way to understand the concept of barycenter is to push a supermarket trolley. If you hold only one corner and push, it will only go in a straight line when you push through the centre of mass. Otherwise a turning force is created. This shows that the mass acts as if it were only in one place.
The center of gravity is a closely related term. It is the point where an object can be balanced perfectly by a sharp point holding it up under the object. It is the center of mass acting in a gravitational field.