There are four different concepts here that are commonly confused, because they are not taught properly in high-school and 1st-2nd year elementary physics courses. There simply isn't time, and later in specialized courses there isn't room:
(1) The Centre of Mass. This concept is used most often in classical mechanics problems. It has a clear unambiguous mathematical definition, but has no physical counterpart or concrete meaning, aside from being a convenient way of managing a system of particles. It is the arithmetic mean of position for each particle in a system weighted by each particle's mass. (for charges weighting is not necessary, so there is no corresponding concept in electrostatics.)
(2) The Geometric Centre. A similar idea, mostly applicable to regular shapes having some form of symmetry, and meant to represent the mean position of a volume in space.
(3) The Centre of Gravity. A point with behavioral properties that help to describe the conditions of balance of a rigid body under the influence of a uniform (or zero) gravitational field. Essentially the opposing axis through a balancing point on a rigid object must pass through this internal point.
(4) The Equivalent Point-Mass. The position in space one must place a point-mass of the exact same mass as the system to be replaced in order to exert the exact same force in magnitude and direction upon another external test-mass.
A brief discussion of the similarities and differences is in order.
(a) For regular shapes of uniform density, or symmetric arrangements of discrete components in a system, the Centre of Mass often coincides with the Geometric Centre, making it easy to locate without having to average all the particles in a system.
(b) The Centre of Mass is used as the common or best estimate of the location of the rest frame for rigid bodies.
(c) The Centre of Mass is useful for studying rotation, since a system of particles (rigid or not) which is free of external forces or in a uniform field will rotate about this point if it contains any stored rotational energy. This is due to the Conservation of momentum and energy.
(d)The Centre of Mass is expected to follow trajectories as if all the mass were concentrated at that point in a rigid body (ignoring things like wind resistance etc.) For example, a spinning hammer may appear to move irregularly, but if the Centre of Mass is followed, this point will move in a straight line through space or trace a parabola if thrown near the earth's surface, obeying Newton's laws of motion for point-masses.
(e) The Centre of Mass is acknowledged to be only an approximation when dealing with gravitational and electrostatic fields. This is because systems that don't have uniform geometric symmetry don't have distributions of mass that exert uniform forces on arbitrarily located external bodies, even when they have uniform densities.
(f) A Key point is that the Equivalent Point Mass location (EPM)is only definable from the viewpoint of a test-mass with a fixed and defined location. If the test-mass is moved, the EPM also moves, when the object under consideration is not uniform in density or radially symmetric in all directions.
(g) An important concept to come away with here is that an irregular object looks different to all other parties and locations in space, even if those objects are point-masses, and so the EPM is in a different location for each observer under the influence of the gravitational force from the object. There is no unique EPM for an object.
