Sharp from here to the horizon
Scenery photographers want a lot of the scene sharp, from a rock a few meters away all the way to distant mountains. Hyperfocal distance is the focus setting that makes that happen. Focus there and everything from half that distance to infinity looks acceptably sharp.
Enter your focal length, aperture and sensor type, and the calculator gives the hyperfocal distance and the near limit, in meters or feet.
The formula, simply
H equals focal length squared, divided by the aperture number times the circle of confusion. The near limit of sharpness is half of H. Focus at H and the sharp zone runs from H/2 out to infinity.
Say H works out to 4 meters. Focus at 4 meters and things from 2 meters to the horizon are in acceptable focus.
What changes it
Aperture. A smaller opening, like f/11, brings H closer to you and extends the zone. Open it up to f/2.8 and H moves far away. A nine-stop table shows the numbers side by side so you can pick the aperture that fits the scene.
Focal length. Wider lenses have far shorter hyperfocal distances. At 16mm, you might focus just over a meter away. At 200mm, H can be hundreds of meters.
Sensor size. Smaller sensors use a smaller circle of confusion, which changes the answer. Pick your camera format so the math matches.
What is the circle of confusion?
It's the largest blur spot you'd still call sharp when viewing a print at a normal size. A bigger sensor allows a bigger value. It's a judgment, not a physical law. If you plan to print huge or crop a lot, use a stricter value.
In the field
Set your aperture, look up H, then turn the focus ring to that distance. Many modern lenses have no distance scale, so you may need to focus on an object at that distance, or use live view and pace it out. Stop down a little extra for a safety margin.
When not to use it
Skip hyperfocal focusing when a subject must be critically sharp, like a portrait or a close-up of a flower. The method gives you acceptable sharpness everywhere and perfect sharpness nowhere. For a wide scenic shot with a key foreground element, focus stacking may serve you better.