This may or may not be what you are looking for. I usually see it as three types...mechanical, biological and chemical.
Mechanical is the physical seperation of debri and solid waste out of your tank water. Depending on the filter you have, this is accomplished by sponges, floss, filter pads, screens and polishing media such as on a diatom filter. They can vary in how pourus they are, some letting smaller particles through and you can have filters with several stages, each sucessive one getting finer to catch more particles. These should be regularly cleaned since as they trap debri, the total flow through them will be diminished. The normal recommendation is to rinse them in old tank water, due to some of your bacteria colony will be on them, and for some types of set ups, most of your bacteria will be there.
Chemical filtration is where you have added a medium that is working with your tank water to remove or add items to change your water characteristics. Activated carbon, crushed coral, and peat are some of the most common ones. These items can be in bags, cartridges, pads or media chambers, depending on what type of filter you have.
Biologigal filtration is the bacteria that develop to convert the ammonia to nitrites and nitrites to nitrates. This bacteria needs oxygen and a food supply. While this bacteria will establish on most any surface in your tank or filter, the generally accepted theory is that most of it will be in your filter. The more surface area you have, the more available area you have for a bacteria colony. This surface area is provided by ceramics, beads, substrate, decorations, and even the tank walls themselves. The bacteria grow best where the best conditions for it are.
Depending on your filter style, the above can be achieved in almost limitless combinations and sometimes one form of filtration may actually be accomplishing more than one method of filtration.
I have canisters with floss sleeves for the mechanical, then the water goes thru carbon for chemical and returns to the tank thru bio wheels. Also have canisters with floss sleeves and running thru chambers with nothing but ceramic cylinders and returning to the tank.
HTH