There are a variety of reasons why you may want to make a recording of a karaoke performance. To share with friends, to post on social sites, for audition purposes, or maybe for making a CD. Whatever the reason you can make good quality recording fairly easily.
You can use just about any microphone that will plug into your computer (USB or 1/8" plugin) although a better quality mic will make for better sound quality. Your recordings can be saved as mp3 files which of course can be used to create an audio CD. Once you learn the program and its settings you can make some good quality recordings.
The basic process is to connect a cable from the non-powered "main mix out" of your mixer or powered mixer to the "line in" of your computer's sound card. The cable you use will depend on the connection types. The line in on a sound card is typically a 1/8 inch stereo type. The output from the mixer may be either a 1/4 stereo jack or the dual RCA type.
You can use audio recording software like MP3 Audio Recorder for example. Have the program set up to record "line in" and set it to where you only need to click a button or tap a keyboard key to start recording. You will need to experiment with the record levels until you achieve the best quality recordings. If you save the recordings as mp3 (as opposed to wav) you will save considerable hard drive space.
Making live recordings can be a little difficult for one person. It's best to have one person handle the recording end while another person takes care of starting the music and handing off the microphone.
The wav files you create can be burnt to a disc in the same way you would burn tracks that were ripped from an audio CD. Mp3 files will require software designed to convert them to wav first and then burn them.