In practical geometry a point is anything that can for some purpose be represented by a dot. The thing represented doesn't actually have to be small. In working out the orbits of the planets in the solar system it is often convenient to treat the sun, planets and other astronomical bodies as points. In cosmology we may even represent whole groups of galaxies as points, though this may be stretching the analogy as far as it will go. On smaller scales molecules, atoms, electrons and smaller particles of matter are often treated as points, though here once again we are stretching the analogy rather a lot. Subatomic structures are now often described in terms of other analogies, such as waves, strings or glue. For ease of visualisation we will think of a point as a carefully made small mark on our drawing paper, but it should be borne in mind that this is only one context. For instance a point on a distant advertising hoarding may turn out to be a very large spot when seen close up.
In traditional geometry points are supposed to be of zero width. It follows that we can find an infinite number of points on any line segment, in fact a super-infinite number of points, since given any point we can always find another point closer to it than any other given point. This raises considerable logical problems. More modern developments of geometry tend not to define what points are at all, only the axioms they obey, such as that two points X and Y determine a unique straight line XY and have a fixed distance d(X, Y) between them.
Since lines are seen as paths followed by light one might think that points could be interpreted as photons. Here is a link to a site that tries to answer What is the size of a photon?. There are many discussions of this on the web. The width of a photon could be said to be about a fermi unit (10^−15 m), since that is the distance at which it could be said to "push back".
The Planck length (1.616×10^−35 m) is usually reckoned the smallest length of physical significance. To quote that wikipedia article: "Because the Planck length is so many orders of magnitude smaller than any currently possible measurement, there is currently no way of probing this length scale directly." and "The Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it would become impossible to determine the difference between two locations less than one Planck length apart." In other words the very concept of distance breaks down at such small distances.