Milling machines were actually based on the Lathe (at least according to wikipedia). Shapers were around before that AFAIK. They work like a lathe in that they use a single point tool for cutting. Instead of the part rotating, and the cutter remaining fixed, as in a lathe, the part is held fixed in a shaper, and the cutter moves back and forth across it in a linear fashion. There is typically an auto feed that traverses the cutter across the workpiece a fixed amount after each stroke. Depth adjustment is usually manual.
I have never operated a shaper, but find them fascinating. Lots of videos on youtube of them in action. There is not really anything you can do on a shaper that you can't do on another machine, but tooling is very cheap (hss cutters, vs endmills or broaches). From what I have seen, the surface finish is also remarkably good. Many people like them for cutting dovetails, as they can do it with cheap tooling, as opposed to a dovetail milling cutter, which can be expensive and delicate.