I've seen various clays or clay like products listed with different types of "cut" like mild or aggressive, fine or medium.
What are the uses for these different "cuts" and how would I choose which one to use?
The different "cuts" are like the coarseness of sandpaper – they describe how deeply the material scratches the surface. The more "aggressive" cuts remove material at a faster rate – this means that you can be more productive, but also that you have to be on your toes to ensure that you don't remove too much. The larger scratches will also (probably) require the use of the finer grades of material to polish them out.
As you're learning, start with the finest and work up to more aggressive materials as you learn about how they behave – it will be slower, but less likely that you will learn a lesson that you regret.