If you have a light bulb that has broken off and the base is still in the socket, the first thing you need to do is turn off the power to the light. Turning off the switch is good, but it is best to turn of the whole circuit to be safe, if it is in a lamp make sure to unplug the lamp.
If the glass has separated from the base but is still intact, place a bag around it and break it, leaving just the base in the socket.
To remove the base of the broken light bulb from the socket, you can try using a potato. Slice the end off the potato and slide it over the center glass post of the bulb and twist. If the center post is also broken off, you will need a pair of needle nose pliers to remove the base.
This video will show you both ways to remove the light bulb base from the light socket.
Remember, Safety first, wear gloves and eye protection, as you will be working with glass and metal that can break off, you certainly don’t want that in your eye!