One of the most frustrating experiences in woodworking can be when a screw hole becomes stripped out and will no longer support the fixture, accessory or accent that it was designed to hold. In this event, you can learn how to fix a stripped screw hole using a variety of methods, as there really isn't a "one-size-fits-all" solution to the problem.
The first step is to determine whether a slightly longer or wider-diameter screw will solve the issue. Perhaps just a bit more bite will solve the problem without altering aesthetics. Many times, this isn't a viable option.
Well, if a larger screw won't be appropriate, the next best thing is to try and determine precisely what type of wood you're dealing with. If you're working with pine, fir or other similar softwoods, you may be able to fill the hole with small strips of the same type of softwood. Simply cut off some thin, but still relatively substantial shavings from a piece of stock of the same species. Dip the tips of these pieces into some wood glue and tap them into the hole until the hole is filled. Allow the glue to dry for a few hours, then trim off the excess with a sharp knife or chisel and sand the fill if necessary. Tap a slight pilot hole using an awl into the filled wood and then drive the screw home. Your problem should be solved.
You can take the exact same approach if the stock with the stripped out hole is a hardwood, but try and use pieces of the same species to fill the hole. You definitely don't want to use hardwood filler in softwoods, as the hardwood might split the softwood when the screw is driven in. By the same token, filling with a softwood filler into hardwood may not be durable enough to accept and hold the screw.
Now, if you can't determine the species of stock, or if you don't have any of the same stock to use for filler, what do you do?
Well, there are other methods for filling the hole, which may work with some success. My personal favorite, particularly in softwoods is to tap a few wooden match sticks (with the heads cut off) into the hole. You can use a few drops of wood glue in the hole before filling it with match sticks, but the wood glue isn't absolutely necessary. The match sticks are consistent widths and are thick enough that they shouldn't snap off when driving them into the hole. Hardwood dowels or toothpicks might be other options (again, based on the stock being repaired).
I've heard of woodworkers using epoxy or even "Bondo" to fill such blemishes, but I tend to shy away from these chemical fixes for this problem. The reason is that wood is flexible, and will expand and contract based upon weather conditions. Hardened chemical solutions will not have the pliability of the wood it is filling, and as such, may cause greater problems with cracking or splitting down the line. In cases where the stock expands with the weather, the plug could even fall out. I would have less of a problem using small amounts of epoxy to hold in the wood plugs mentioned in the previous paragraphs, but I'd wait until the epoxy has thoroughly cured before re-seating the screw. If the epoxy grabs the screw before it dries, you may never get the screw out in the event that you need to make other repairs or disassemble the piece later.
One final note: in the stain aisle of your local home center, you'll find plenty of "wood fillers." These products are designed to help fill holes before (or after) applying a finish. They are for aesthetic purposes only, and are not appropriate for fixing stripped screw holes. If you try to use a wood filler for structural purposes, you're likely to be sorely disappointed, and be right back to square one relatively quickly.