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 square and continuously deform it without tearing or gluing, there will always be at least one point that remains in the same position. This theorem is significant in various fields, including mathematics, economics, and game theory. It provides a foundational concept for understanding equilibrium points in systems where entities interact, ensuring that solutions exist under certain conditions.