Screws can be turned too tightly, the head can rust off, rust on the shaft can hold the screw in so tightly that when trying to remove the screw the head breaks. The screw can be drilled out with a drill bit larger than the screw. However, in the case of a door hinge, the larger hole will become too large to reuse for a screw for a new hinge. A tool called a "screw extractor" grabs the screw by creating a new groove in the shaft so the screw can be removed without enlarging the hole.
Put on safety glasses or goggles.
Insert a screw extractor into a drill. Put the drill in the "reverse" setting.
Place the tip of the screw extractor into the centre of the screw with the broken head.
Press the tool into the screw with a broken head to create the groove in the shaft to help remove the screw. Press hard to push the tool into the screw to create the grooves needed to remove the screw.
Apply steady, even pressure while turning the screw with the drill in reverse.
If the screw is rusted, which is typical of older door hinges, try spraying it with a lubricant, such as WD-40, the night before removing the screw to loosen the hold the rust creates on the surrounding surface. Use a soapy cloth to remove the lubricant before using a screw extractor so the tool does not slip on the lubricant.
All tools can break. Always wear eye protection for safety.