Turn the power off to the wiring at the main panel. Use wire strippers to strip off about an inch of insulation from the ends of each wire you will be splicing. Be careful not to damage the wiring while stripping off the insulation.
Splice the two ends of bare wire together by placing them alongside each other and twisting with a pair of electrical pliers. Three or four turns of the pliers makes a sufficient connection.
Attach a wire nut at the point where the two wires twist together. Twist the wire nut until it fits snugly, or until you can't twist it anymore. Make sure the wire nut is the correct size for the gauge of wire you use.
Wrap the wire nut and two wires with electrical tape to secure the connection. Cover any exposed wiring with electrical tape.
Tuck the wires into the junction box or outlet, depending on what type of project you are doing. Install the cover plate. Turn the electrical power back on and test to make sure the splice is good and you are getting power to the appliance or outlet.