Originally Posted by Ishniknork

There's a thing called an easy-out. It's kind of a left-hand twist, tapered removal tool for just that type of thing. They come in different sizes according to the size screw, bolt, or stud you're trying to remove.

You drill a hole in the center of the broken screw with the appropriate size bit (one that's smaller than the diameter of the shaft the threads are on) and screw in the easy-out in counter clockwise. The tapered shaft has left-hand ridges on it that grab the inside of the hole you drilled causing it to dig in so you can remove the screw.

Pep Boys has them, and probably most good auto supply stores. Knowing the size screw that's broken would help you get the right size.