If an ellipse is the set of points whose distances to two foci add up to a constant, a hyperbola is the set of points whose distances to two foci differ by a constant. Fix two points F₁ and F₂ (the foci) on a plane, and gather only the points whose distance difference to those two points always stays the same — you get a curve that splits into two branches, and that's exactly what a hyperbola is.
Translate this definition into an equation, and you get the familiar x²/a² − y²/b² = 1. Here a is the distance from the origin to a vertex, c is the distance from the origin to a focus, and b is defined by the relation c² = a² + b² (note the sign is flipped from an ellipse's c² = a² − b², which is easy to mix up). The difference between the distances to the two foci is exactly 2a, always. On the activity page's "Two-Focus Definition" tab, move the sliders for a, c, and the point's position, and you can confirm for yourself that this difference always stays at 2a.
A hyperbola has a unique property that neither an ellipse nor a parabola has. Look far, far away from the origin, and the two branches of the hyperbola get closer and closer to two straight lines through the origin, y = ±(b/a)x. These two lines are called the asymptotes. The hyperbola gets endlessly closer to its asymptotes but never actually meets them — no matter how far out you go, the gap between them never quite reaches zero. On the "Asymptotes" tab, change a and b and watch how the larger b gets relative to a, the steeper the asymptotes stand and the faster the hyperbola opens up.
The hyperbola's "constant difference between distances to two foci" property is also put to real use in location-finding technology. A classic example is the LORAN navigation system once used by ships at sea. Two fixed transmitter towers send out signals at the same moment, and a ship measures the time difference between when the two signals arrive. Knowing that time difference tells you the difference in distance to the two towers, and the set of points with a constant distance difference traces out exactly a hyperbola. So the ship can tell "I'm somewhere on this hyperbola." Add a second pair of towers and you get another hyperbola, and the point where the two hyperbolas intersect is the ship's exact location. GPS uses a similar idea: by comparing the arrival-time differences of signals from multiple satellites, it produces several hyperbolas (more precisely, hyperboloids, since it's three-dimensional), and calculates your position from where they intersect.
To sum up: a hyperbola starts from one simple rule — "the difference between distances to two foci stays constant" — and from there it's expressed by the equation x²/a² − y²/b² = 1, it comes with asymptotes it endlessly approaches but never touches, and it forms the mathematical backbone of navigation systems that pinpoint location from signal-arrival time differences. Now that you've looked at the ellipse, the parabola, and the hyperbola, look into what "conic sections" means too — the fact that all three of these curves are just cross-sections of a cone, sliced at different angles.