A hyperbola is a type of smooth curve in mathematics defined as the set of points where the difference of the distances to two fixed points, called foci, is constant. It consists of two separate branches that open away from each other. Hyperbolas are one of the conic sections, along with ellipses and parabolas, and can be represented by equations in a Cartesian coordinate system. They have various applications in physics, engineering, and navigation, particularly in describing certain types of motion and trajectories.