The best thing to do is go to radio shack and get a crimp type butt splice. You can buy a cheap inexpensive crimp tool and they will be happy to show you how to use it.
The other alternative is to use a wire nut. By stripping about a 1/2" of insulation off each end of the wire you can then twist the two ends together then cut of the excess bare wire. You should have about a 1/4" to 3/8" bare wire now showing. Then use a blue wire nut which is for smaller gauge wire like #16-#22awg. If the wire is larger like #12-#14 then use a yellow wire nut.