When changing a light bulb, occasionally the glass portion of the bulb breaks loose, leaving the base in the fixture, often with jagged edges. Try this trick. Go get a raw potato, shut off the power, insert the potato into the broken base, and use that to twist it out.
If this doesn't work, apply a spray-on penetrating oil and try again. As a last resort, try using long-nose pliers. Wear safety glasses and be careful of broken glass.