Point
A point is an exact location in space with no size, length, width, or depth, represented by a dot and named with a capital letter.
Definition
A point is an exact location in space with no size at all, no length, width, or depth. We draw it as a tiny dot and name it with a capital letter, like point $A$. As a fundamental (undefined) term in geometry, a point is used to define every other figure: lines, shapes, and solids are all built from points. In analytic geometry a point in $\mathbb{R}^n$ is an ordered $n$-tuple of real numbers, and in topology it is simply an element of a set on which a topology is defined.
Example
The tip of a sharpened pencil, or the dot marking your city on a map, is close to a point: it shows an exact location but has no real size. Points $A$ and $B$ are two distinct locations in a plane; a line passes through both, and a segment connects them, with coordinates like $(3, 5)$ naming a point on the coordinate plane. In $\mathbb{R}^2$, a point $P = (x, y)$ identifies a unique location, and the set of all points equidistant from a fixed point defines a circle.
Key Insight
Everything in geometry starts with points, making them the most basic building block of the subject. Points are called "undefined terms" because they are too basic to define using simpler ideas, we accept what a point is intuitively and build every other definition on top of it. Euclid's attempted definition ("that which has no part") is circular, so modern axiomatic systems (Hilbert, Tarski) instead treat point as a primitive object satisfying stated axioms, avoiding that circularity.