The Brouwer fixed-point theorem states that any continuous function mapping a compact convex set to itself has at least one fixed point. This means that if you take a shape like a disk or a ball and continuously deform it without tearing or gluing, there will always be at least one point that remains in the same position. For example, if you imagine pushing down on a rubber disk while keeping it flat, there will be a point on the disk that does not move. This theorem is fundamental in various fields, including mathematics, economics, and game theory, as it helps in understanding equilibrium states.