Beware the broken light bulb
Removing a broken light bulb from a screw-shell lamp holder can be dangerous if you don't solve two problems: First, how do you know the electricity is really off if the fixture or lamp holder is controlled by a simple on-off switch (or a pull chain)? Second, how do you get the broken light bulb base out of the lamp holder without cutting your hand or damaging the inside of the lamp holder?
Removing a broken light bulb
If you know which circuit in your electrical panel controls the light fixture's power, turn off the circuit breaker or remove the fuse. Then, verify that the correct circuit is really off. Check for the presence of voltage in and around the lamp holder with a tool such as a non-contact voltage detector.
Still in doubt? To be absolutely sure you'll be safe, turn off the main breaker at the electrical panel. After you are sure the electricity is off, use a small potato for a unique, effective way to take the light bulb out of a lamp holder. Taper-cut the potato's tip so it will fit into the lamp holder. Wearing gloves and safety glasses, break off any glass bulb parts that will interfere with the potato, then insert the spud and twist it counterclockwise.
Want a more conventional approach? Use needle-nose pliers to remove the bulb. Firmly grab the bulb's metal base and avoid damaging the light fixture's metal screw-shell lamp holder. When replacing the bulb, consider installing a shield around the globe if the light fixture is in a vulnerable location.