A regulator, very simply, is the thing that goes on the air tank and allows you to breath the compressed air at a pressure that won't blow out your lungs. It has two pieces, or stages, that are connected by a rubber hose. The first stage screws to the vave on the tank and the second stage has a mouthpiece and goes in your mouth.
Scuba tanks are filled to a pressure of 3000 pounds per square inch and would explode the little sacs in your lungs if you breathed off of them directly. The first stage has a demand valve that steps the pressure down to about 125-140 psi above the ambient pressure around you. Still too high to hold the mouthpiece in your mouth. The second stage has another demand valve that steps the pressure down even further and gives you just enough pressure and air to breath comfortably.
I'm sure you'll get guys on here that can explain it better but that's more or less what one is. If you are scuba diving, you'll be using a regulator.