Introduction: The center of mass is the point from which, for calculational purposes, we can consider all of the mass to be concentrated. Physically, it's the point from which the object can be suspended without it rotating.
This implies that torques are a consideration; "no rotation" when it is suspended means that the sum of the torques must be zero.
Consider a child's "seesaw". Two kids sit on the ends of a board. The board rests on a fulcrum (pivot) which is not at the center of the board. If the children are the right weights, and are positioned correctly, they will not rotate (unless they force rotation by pushing with their legs). The center of mass of the system consisting of the board and the two kids is at the fulcrum. If you were given the masses and the positions of the two children, you could find (using torques) the place where the fulcrum would have to be placed.
Let's look at this using mathematics...please go to Example 1.