If u watch a live rap performance, sometimes MCs freestyling on a beat from song that has no chorus nor singer's voice on. I'm sure it's the DJs that gives the beat, how do DJs do that, are they playing a record that nothing but a beat on or is that a function of a mixer?

Its just a pre-recorded beat on the record or the cd or a part of a track which is made into a streemless loop. You can download some of those from kazaa.

Hope this helps
Cheers
MJ