In the Tipler physics manual there was a nice pic of two sea gull models hanging from a thread.
One of them had its mass evenly distributed, and its was hanging from its CG, in the middle of it. The other sea gull had its mass concentrated in the tip of one wing. It hanged from its CG which was in the wing, not in the center of the shape.
Distribution of mass will determine the CG. In fact, it determines the center of masses. For small bodies, this and the CG are coincident. For superlarge objects such as the moon, the CG and the center of masses will be close but not coincident.