There are a couple of ways to fill countersunk screw holes. The easiest way is to purchase buttons, which are available at most home improvement centers or woodworking supply shops. Simply put a little glue on the button to hold it in place, position it into the hole and give a tap with a hammer. This type of plug does not actually hide the hole, but instead makes it decorative.
If you would prefer to make the hole disappear completely, begin with a piece of scrap wood that closely matches the wood that the hole is in. Use a portable drill press to hold a power drill in position and use a plug cutter attachment to match the hole size on the drill. Drill down into the block of wood. Pop the plug out of the wood, put a little glue it on the base of the plug and push the plug in position.
Align the grain of the plug with the grain of the wood, and tap it in place with a hammer. Allow it to dry for a few minutes, chisel and trim off the excess and then sand the surface flush.