There are three main categories of filtration - Mechanical, Chemical, and Biological. Mechanical filtration traps and removes suspended matter, i.e., waste products, and prevents chemical and biological filters from becoming clogged with debris. The most common form is spun nylon floss. Chemical filtration is usually based on active carbon, which has the capacity to absorb the ions from dissolved waste onto its large surface area. It also harbours some of the bacteria responsible for breaking down potentially harmful ammonia and nitrites.
Biological filtration comes in a variety of forms.
They cleanse the water by harnessing the biochemical properties of living bacteria and break down waste. I have found advanced forms better, such as an above-water trickle filter, which allows a free flow of atmospheric oxygen to reach the bacteria. A much larger colonization of bacteria can be maintained by using a similarly sized filter bed below water. I am a big believer in using Bioballs. These plastic balls, which look rather like stickle bricks, have a huge surface area, which means they are effective against even more bacteria.