A folding pipe is defined by having the ability to fold on itself. Intuitive enough. Mechanisms for the folding ability range from a rotating stem to magnetism. Folding pipes are usually very compact and optimized for smoking on the go; they range down to less than 3" in the closed position. The selling point of a folding pipe is that when closed, the bowl is covered and concealed, allowing it to be loaded, then dropped in a pocket and carried in relative safety. Most designs have no resemblance to traditional smoking devices when closed, too. Many designs also include an extra stash to contain another bowl's worth or more within the structure of the pipe. A folding pipe makes a very simple, convenient, stealthy pipe that can be dropped or thrown without even losing its bowl, much less breaking.
Due to the engineering problems related to building moving parts out of glass, glass folders are rare to nonexistent. If you want glass, you don't want a folding pipe. These are not that great for home use, with the possible exception of nosy-parent situations. It is also possible for some hinge mechanisms to wear out or lose parts, defeating the point of a folder or possibly rendering it useless.
Monkey Pipe The Monkey Pipe is probably the best known folder on the market. It sells for around $25. Its construction is mostly wood with a few metal moving parts, and it measures 2.125" X 0.5" X 1" when folded. There is a stainless steel insert in the bottom that collects resin and cools the smoke a little, and it can be removed for cleaning. The stash on the higher-quality model holds about a bowl and a half. It doesn't leak air at all and hits pretty smooth for its size.