When the screw driver end of the screw is stripped, and you can't get a hold of the head with a pair of Vise-Grips, then you need to resort to some destructive removal. The goal is to remove the dead screw, without damaging the material around it.
Create a Slot
You can often use a hack saw, or a Dremel grinding wheel, to cut a slot in the top of the screw head. This may allow you to remove the screw with a flat screwdriver. You remember what a flat screwdriver is. It is that finger gouging thing that thry still use in the US to drive screws since they haven't discovered the Robertson square socket yet.
Use an Easy Out
If that won't work, then you need to use an "easy out". This is a pyramid looking bit of hardened steel (that comes in various sizes) that has a backwards spiral winding around it.
First you drill a hole into the head of the screw. You don't want to go too far, nor too large or you will simply drill of the screw head and loose the rest.
Then you use the easy-out with a wrench, or a tap handle, and screw it into the hole turning it to the left, the direction for "un-screwing" the screw. As you turn, that backwards spiral will dig into the screw and lock tighter and tighter, transferring your twist to the screw itself, which will begin to unscrew.
This gadget was actually invented to remove machine bolts that lost their hex heads, but with care the smaller ones will remove wood screws too. You can find them at your local hardware store.