An electrical plug prong that breaks off in an outlet creates two problems, one with the plug, the other with the outlet. The first rule of addressing such a situation is basic: turn off the electricity to that outlet immediately, even if the broken prong is the grounding prong (the bottom of three) on the cord. The grounding prong is the one most subject to damage but sometimes one of the power prongs may break; the wider of the two top prongs carries the current, the other handles the neutral or return wire that completes the circuit.
Turn off the circuit breaker to the outlet. Test it with a circuit tester to make sure the power is off. Inspect the broken prong to see how deeply it is embedded in the plug. If it is sticking out enough, grab it with a pair of needle nose pliers and pull it straight out. Put the pliers on the sides of the prong, which has a flat top and a rounded bottom.
Remove the outlet if the prong will not pull out. Take off the cover face plate with a screwdriver -- it usually is fastened with one screw in the center -- and remove the mounting screws at the top and bottom of the outlet. Pull the outlet straight out of the wall. See if there is a way to push the prong out from the back; most plugs will be sealed from the back, however.
Replace the outlet if the prong won't come out. Attach a new outlet to the same wires as the old one, if it uses screw connections; if it had push connections, cut the wires, strip the ends with a wire stripper, bend new loops with needle nose pliers and attach it with the screws, brass to black wire, silver to white wire and green to bare ground wire.
Change the defective plug. Never use a plug with any damaged prong or with the ground prong broken off. Cut off the old plug, use wire strippers to strip about 1/2-inch of insulation off each wire and connect the wires to the new plug. Connections vary in method and layout, but it is black wire to brass terminal, white wire to silver and bare wire to green.