How to remove harden screws and try preserve the two metals that the screw bind together
There is a product called an 'Ezy-out' (or some variation) and other brands of similar products which are basically a reverse thread, tapered, coarse pitch tap or drill-bit. You can use these by hand or in a drill - in reverse-, once it bites into the screw it will begin driving itself in further or unscrewing the stuck screw.

What size and material is the stuck screw?

Liverally coating the screw in a penetrating lubricant like WD-40, applying heat to the screw to make it expand or tapping it with a hammer while trying to remove it will all help to some dregree.