In general, there are several different ways to cover screw holes in wood. If you plan to paint the wood, you can cover them with wood filler. If you do not plan to cover the wood with paint, you have two methods you can use. Both involve using a wood dowel plug. In total, three processes are covered that will allow you to discreetly cover a screw hole in just about any wooden surface.
Painted wood surface
Countersink the screws into the surface of the wood with a screwdriver. You do not have to sink the screws deep into the wood's surface.
Apply a small amount of wood filler onto a drywall spatula and cover the hole with wood filler. Hold the blade at a 45 degree angle to the wood surface, and scrape firmly and slowly across.
Wait 24 hours and then sand the surface of the wood with a sanding block until the wood filler is level with the wood surface.
Remove the screw from the hole with a screwdriver. Drill a larger hole between 1/4-inch and 1/2-inch into the surface of the wood with a drill. This creates a countersink hole. Reinsert the screw. If necessary, use a shorter screw. If you already have countersunk screw holes, skip this step.
Cut wooden plugs or dowels out of the same type of wood your project. You can purchase pre-cut wooden plugs manufactured with a domed top. These plugs are designed to sit above the wood's surface. You can also purchase pre-cut plugs designed for a flush installation. If needed, you can purchase a plug cutting bit and cut your own plugs on a drill press. Whether you purchase the plugs or make your own, make sure the size of the plug matches the size of the hole your screw rests in.
Place a drop of wood glue into the hole the screw rests in and place the end of the plug in the hole. Tap the plug into the hole with a rubber mallet. If you used domed plugs and you intend to let them protrude from the surface of the wood, you can wipe the excess glue off the wood surface with a moist rag and stop. Your install is complete. If you want a flush finish with the wood surface, wipe off the excess glue, wait 24 hours for the glue to dry.
Place a flat saw against the wood surface and cut the plug flush with the wood surface. A flat saw is flexible and will rise along with wood's surface without damaging the wood.
Sand the wood surface with a sanding block to blend the plug in with the wood's surface.