Moonshine is basically an alcohol made from fermented yeast. It is not stored in aging barrels, and often has an off-putting flavor to it. The reason that it is illegal is because it is an undistilled alcohol. The history of moonshine is available at http://www.ask.com/wiki/Moonshine.
Moonshine whiskey is made of a mash which is made from corn. It is traditionally made in copper stills, but can also be made in a pressure cooker. Moonshine is illegal and has an alcohol content of at least 80%, making it somewhat unsafe to drink.