Welcome on board AQ.
Basically there are 3 main purposes for using ceramic rings and bio balls.
1. Provide settling area for benefitial bacteria for breaking down of toxic stuff like amonia.The porous or "rough" ceramic rings provide the most surface area for bacteria to settle. Bacteria not only can settle on the top surface but also in the pores of the ceramic rings. The "smooth" type of ceramic rings have less pores and therefore less surface area and not so effective compared to the porous type. Bio balls also have lesser surface area compared to the porous ceramic rings. However, they are less susceptible from clogging and are usually us in more heavy-duty filtration, e.g. pond filters. This is usually the last stage or near the last stage of the filtration.
2. Distributing the water flow over the other filtermedia. Water flow is general faster in the centre compared to the flow near the wall. With ceramic ring or balls in the begining stage of the filtration, the flow pattern is broken up and you have a more even flow throughout the filter media. Usually, the non-porous ceramic rings are used for this purpose typically in a canister filter. Bio-balls, because of its size, can be used in larger filter system like the pond filter or sump filter.
3. Provide large contact area between the air and water. This is for the trickle type filter or the wet-dry filter, but not applicable for canister filter. Bacteria needs oxygen to breakdown the waste in the water. This filter system relies on large surface area of contact between water and air to aerate the water so that the bacteria can have sufficient oxygen for the waste breakdown. Usually bio-balls are used for this purpose. As water trickle through the balls, large contact area is created between air and water.