a point
Think of the tip of a sharpened pencil pressed lightly onto paper, or a single star in the night sky: it marks a place but takes up no room. That is what a point is meant to capture — a position, and nothing else. It has no length, no width, no thickness; you can name it but you can never really fill it in.
In Euclidean geometry a point is one of the undefined terms — we deliberately do not give it a definition. Instead we describe how points behave through the axioms: through any two points there is exactly one line, a line is made of infinitely many points, and so on. We label points with capital letters, A, B, C, and draw a tiny dot to suggest one, but the dot on the page is only a picture; the true point has zero size. In coordinate geometry a point in the plane is named by an ordered pair (x, y), and in space by a triple (x, y, z).
A common misconception is that a point 'has no definition because it is too obvious to need one'. The real reason is logical: any system that proves things must start from primitive notions it does not prove and does not define, or it would chase its own tail forever. A point is whatever satisfies the axioms — in one model it is a dot on a plane, in another a pair of numbers, in another something stranger. The geometry does not care what a point 'really is', only how points relate.
On a map, the single dot marking a city is treated as a point: its position matters, its size on the page does not. Two such dots, say A for one town and B for another, determine exactly one straight road line AB between them.
A point captures position alone; its drawn size is just an aid to the eye.
The dot you draw has real width, but the point it stands for has none — never reason from the size of the mark on the page.