The center-of-gravity is the point that moves as though all the weight is concentrated there.

In a uniform gravitational field, weight and mass are proportional to each other and the center-of-mass and the center-of-gravity are the same point.

They're different if the gravitational field is non-uniform. We're not going to worry about that.

The center-of-gravity can be determined by hanging an object from a support. At equilibrium, the center-of-gravity will always hang directly below the support.