Point mass
A point mass is an object with mass but negligible volume. Two bodies may be treated as point masses when their dimensions are negligible compared with their separation.
What this actually means
This looks like a throwaway definition until a question asks 'state one assumption made in your calculation'. The answer is usually that the bodies are treated as point masses, because their sizes are tiny next to the distance between them.
There is a second, stronger result you are allowed to use without proof: outside a uniform sphere, the field is identical to that of a point mass of the same total mass located at the centre. That is why you can apply F = GMm/r² to whole planets with r measured centre to centre.
This is also why the Earth-Moon force calculation works with r = 3.8 × 10⁸ m even though both bodies are thousands of kilometres across. Their radii are a fraction of a percent of the separation.
Where it breaks: a person standing on the Earth's surface is not far from the Earth compared with the Earth's radius, so you use r = R (the radius), not r = 0. Every year someone puts a tiny r in and gets an absurd answer.
Using surface-to-surface distance instead of centre-to-centre separation.
Prove it — watch it be true
- Open the g vs r graph, which is plotted both inside and outside the planet.
- Confirm that outside the surface the curve is exactly the point-mass GM/r² shape.
- Move the probe inside the planet and see the curve switch to a straight line rising from zero, proving the point-mass result only holds outside.