I managed to build this lathe in about a week, with not much more than a cordless drill, a drill press, a jigsaw, and assorted hand tools. I hope that I have documented my project here in an understandable way.
Warning: This is a powerful device designed to spin stuff quickly. I take no responsibility for anything you do. Don't try this unless you have at least a little bit of experience with tools. And wear safety glasses when using it because particles fly around.
Step 1: Decisions
The first thing you have to decide is what kind of lathe you want. Either to work with metal or wood. A wood lathe requires a less powerful motor and not as close tolerances. Also a wood lathe does not need the complicated tool rest that a metal lathe has. For the first version of my lathe, I decided to just stick with wood and see if I could come up with something that actually worked.
The next thing to decide is size. I would highly recommend not going too overboard... tree size logs on a wood lathe and 50 pound steel bars on a metal lathe are best left to professionals I think. I decided to try to make a wood lathe for pieces up to 4 inches in diameter and about 30 inches long, although I will not be trying something that big until I get more practice with small items, like tops, chess pieces, other little toys. But I figured that I had a pretty powerful motor sitting around, so I might as well make it big enough to handle large salt shakers and chair legs so in the future I could do large things.