The light bulb broke off but the metal portion remains in the holder. What is the best or safest way to remove this bulb from the holder?
Make sure there is no power going to the light, and use a potato. Cut it in half, jam it in and twist(unscrew) to remove.
If you can't be sure which switch position turns off the socket (say, with a double switch that lets you control the light from two locations), locate and shut off the breaker for that circuit, or remove its fuse -- or if a lamp, unplug the cord.
The potato trick works well as long as there's glass left on the socket (the potato needs to be big enough to catch on the broken glass attached to the base). If there's no glass left, but the central insulator post is intact, I've successfully removed the broken bulb by carefully gripping the internal glass insulator and using it to twist out the base (gloves highly recommended to protect your fingers in case the insulator breaks), or used small-nosed pliers to grip the edge of the base (which protrudes a few millimeters above the socket) and unscrew the base.
Isolate the power to the circuit at the fuse box/consumer unit/breaker panel/whatever your country calls it first. DO NOT rely on lightswitches to isolate the power. If you know what other things are on the same circuit then you can use them to double check that the circuit is off. If you have a non-contact voltage detector you could use that to double check.
Make sure the circuit will not be turned back on. In a commercial situation I would want to see a lockout. With my own family I consider telling them not to turn the circuit back on to be sufficient.
Then use pliers to remove the remains of the bulb from the socket. Ideally these pliers would have approved insulated handles but at the very least you should ensure you don't touch any metal parts of the pliers during the work.
A good rule of thumb in life is to try and make sure that wherever possible you are at least two failures or screwups away from danger. Hence why you should isolate the power AND use insulated tools.