The Faraday disk generator is an oddity of electrodynamics. It has no stator component (the whole thing turns) and it is hard to understand where the dynamo action comes from (when you put a load on a generator you should feel back-torque). This has led many quacks to believe that the FDG is a source of free energy, which it in fact is not. It does happen to be a mechanically simple way of generating a heckuva lot of DC current, making it ideal for applications like building railguns.
To build an FDG you bind a magnet to the center of a large, non-ferrous conducting disk and spin it. A potential difference will form between the edge of the magnet and the edge of the disk, which you can draw off with brush contacts (or better, liquid mercury contacts). And it's that simple.
So, why does the FDG work? The fundamental principle is that when you rotate a magnet around its axis the magnetic field remains stationary. This is why no stator is required; it makes no difference if the magnet is attached to the rotor or not. Mechanical convenience and physical coolness dictate thus dictate that the magnet should spin.
Variants are possible, such as a metal tube with magnets on the ends, oriented oppositely so that the field only crosses the tube in one orientation. In this case the voltage arises between the ends of the tube. This configuration has the advantage that the contacts are mechanically identical.