Unfortunately, I'm too poor to afford any new chairs to make over, so the pictures on this one are limited because I didn't plan to make a how-to with them. Nevertheless, enjoy. :)
Get an ugly chair.
Flip it around and unscrew the seat from the underside of the chair. This is what it looks like after.
Use pliers (and screwdriver, if you need it to get under the staples) to pull out any tacks and staples from the edges of the cloth. In some chairs -- mostly armchairs -- the stuffing is shaped mainly by the cloth, so if you're cautious you can skip this step and just cover the existing seat with your choice of cloth.
Now take the cloth that was used for the chair and trace around it on the cloth you plan to use to make a pattern (or just measure the chair cushion dimensions and make it that way). Sorry, doesn't have the picture for this one.
(Or this one)
Now fold your new pattern around the cushion, trying to not get too many wrinkles, and use your staple gun to attach it. If the cushion is square, the cloth at the stapling side of it should fold over to look like the corners of a picture frame.
Screw the cushion back to your chair, and voila!