Undergravel filters are the most common type of biological filter and work by pulling water down through a bed of gravel at the bottom of the aquarium and pulling it up through the uplift tubes. Reverse flow undergravel filters are set up to pull water from the top of the aquarium by pushing it down the uplift tubes and up through the gravel. Reverse flow filters are slightly more efficient since they take water from the top of the aquarium which contains more oxygen.
Since nitrification is more efficient in the presence of oxygen and the air contains about 20% oxygen, as compared to about 7% oxygen for water, trickle filters were developed in the 1980's which increased the efficiency of biological filters dramatically. They place the bacterial growing medium, filter balls with large surface areas, in the air (usually outside the aquarium) and trickle the water to be filtered over them. Many kinds, sizes, and shapes of trickle filters have been in use since they were introduced. Although many trickle filters are external devices used in a sump or hang on the back type of filters, several manufacturers including the "Sea Clear System II" aquarium contain trickle filters built into the back of the aquarium itself. The problem with trickle filters is they usually produce high levels of nitrate, and so are not often used in reef aquariums or other situations where nitrates are undesireable.

Sponge filters work by passing water over a highly porous sponge where bacteria are allowed to grow. They are almost purely biological with some mechanical filtration provided by the sponge itself. The sponge should not be cleaned or allowed to dry out because this will kill the nitrifying bacteria. Sponge filters are good for breeding situations where very small fish are at risk of being sucked up by the strong intake currents of canister filters or overflows. They are also good for fish like bubble-eyed goldfish that may be injured by strong intake currents of power or canister filters. Sponge filters can be powerd by air pumps or small water pumps.

Canister filters have a containers or baskets inside the filter for various filter media. For biological filtration you would fill it with some sort of biological media, such as bio balls. This media will either come with the filter or can be purchased separately. Once the bacteria colony has been established on this media you will not need to remove it or clean it.
Standard Power filters use disposable cartridges that have a floss pad surrounding them. This pad is primarily for mechanical filtration but will also house nitrifying bacteria. The drawback to this as a primary biological filter is when you dispose of the cartridge, you dispose of the bacteria colony as well. Today there are more advanced power filters that include a separate area for biological media. These are usually in the form of a compartment in the filter filled with biological media or an attached external 'bio wheel', a pleated wheel that the water flows over as it is leaving the filter and returning to the aquarium.

Regards Gary