Anyone who has seen "M.A.S.H." or a good moonshine and bootlegging movie like "Thunder Road", has seen a home distillery, or simply a "still." The simplest version of still to build at home is the pot still, which is actually still used to make most scotch and virtually all Irish whiskey today. The design is very simple, and the principle has changed very little over the last several decades.
Purchase or convert a sealed copper boiling pot. Remember that this pot is where your substance to be distilled goes. In the case of whiskey, this would be a grain mash.The size of it will determine how much you can distil at once. If you are making your own pot, you will find that the simplest design involves a clamp-and-seal lid, much like that of a kitchen pressure cooker. Either way, you will need to closely monitor the temperature of the heat source, to make sure it is boiling the alcohol and not the water.
Consider how you intend to heat your pot. The simplest version of a copper pot still involves putting the boiling pot on stone blocks or metal stilts, and burning a fire underneath it to provide heat. This is still common practice for moonshiners. Industrial stills involve circulating steam through the bottom of the copper pot in pipes. You might need to do this if you plan to run your still indoors, but overall your heating system need only be as complicated as you make it.
Run a condensation coil from the boiling pot to the collecting pot. This is a tube is typically made of either copper or glass. These tubes are where the alcohol will condense and collect, and it should slope downwards to the collecting pot, so that gravity will do the work of drawing it into the collecting pot.
Purchase or convert a collecting pot. This also needs to be sealed, and will fit into the other end of the condensation coil. The only added consideration for the collecting pot is how big you want to make it. These are usually smaller than the boiling pot, but if you want to put multiple distillations into the same pot without having to disconnect and empty it, it ought to be as big or bigger. However, a big pot may be too big to move when full, and will need a faucet installed on the bottom.
A still works by taking advantage of the different boiling points for water and alcohol. At sea level, water boils at 212º F, while alcohol boils at 172º F. Therefore, if you heat a alcohol-containing grain mash enough to boil the alcohol, but not the water, the alcohol vaporises, rises out of the pot and into the condensation coil, where it cools down and turns back into liquid, and then drips into the collecting pot.
Building a home distillery from scratch is beyond the capabilities of a person who does not already have an intermediate grasp of plumbing and welding. If you do not already have those skills, do not consider fashioning or converting copper scraps into a pot still. Instead, purchase a kit and assemble one. Distilling is dangerous business, and improperly distilled alcohol can be poisonous.