All-purpose drywall mud is the type of mud that is the most flexible. The option exists to use all-purpose mud for every portion of the drywall process. This type of mud comes already mixed in a bucket, so you just dig in and start working. All-purpose mud can be used for taping drywall and for completing the secondary and final coats. Some people even use it to add texture to the wall. Although it can be used for any application, it doesn't always work as well as specific products designed for specific portions of the process.
Topping mud comes in the same type of packaging as all-purpose compound, but it holds a few advantages over all-purpose for putting on top coats after the initial taping has been completed. First, it shrinks less than the all-purpose product, meaning you get better coverage with your upper coats. Secondly, it also ends up being much easier to sand after it has dried. Overall, this makes it easier to work with when doing the final coat on a wall. It is also lighter in color than other types of mud.
Taping mud is used for taping, and sometimes for the second coat as well. Taping mud is sometimes a better option than all-purpose mud for the base layer on a wall because of the added strength that it provides. That strength means that the joints and mudded areas of a wall filled with taping mud are less prone to cracking than one where only all-purpose mud was used.
Quick-drying mud comes in a mix that you must combine with water before use. After applying it, this type of mud dries very quickly. This doesn't give you much time to work with it, making it better for use on small repair jobs. It is also better suited for use by experienced drywall workers, as there is little margin for error with the rate at which the compound dries. It is, however, very effective when you need to complete a quick patch job.