As per usual, I will give the classical, textbook, view; an important starting point.

The *Centre of Mass* of an object or a set of objects is an actually defined point.

Assume, for simplicity, that there are two masses, small enough to regard as points.

For two masses M1 and M2, the CM lies on a line between the two masses at distances D1 and D2, respectively, from the masses such that

M1.D1 = M2.D2

That equation is the same as you use for working out 'moments' with levers in school, remember. When the fulcrum is in the right place, the moments are balanced (equal and opposite).

The CM is the point where you could place a mass of M1+M2 and, for a **distant** third, point, mass, the gravitational attraction would be the same as for the two separate masses.

You can extend this to any number of points or the distributed mass of a real sized object if your maths is up to it.

The CM is often called the Centre of Gravity because, for objects on the Earth's surface, the CM is where Earth's gravity can be thought of as acting. CG isn't really the proper term to use in other circumstances, though.

Imagine the Earth and the Moon, as a pair of masses. If you are on the Earth, the place where the gravity of the two objects acts on you (the CG) is, definitely, pretty much downwards - near the centre of the Earth. If, however, you are on the surface of the Moon, the place that the effective gravity of the two masses acts on you (the CG, again) is 'downwards' -i.e. near the centre of the Moon.

So this 'centre of gravity' is in different places, depending on where you are looking from; it is dominated by the closer masses. Hence, it is a rather woolly term to use.

Why does the CG of an object appear to be the same as the CM on Earth? It's because the Earth is big enough to seem flat for most objects and the gravitational 'lines of force' are virtually, parallel ('vertical') - as if the Earth were at infinity.