Most aquarium keepers rely on a combination of three different filtration methods to keep the water quality up in their aquarium: mechanical filtration, biological filtration and chemical filtration. Mechanical filtration will remove larger debris from the water, biological filtration is performed by bacteria that convert dangerous waste products into less harmful compounds, and chemical filtration is typically performed by activated carbon that will bind toxic compounds, traces of medication etc.
Several types of filtration can take place in the same piece of equipment. When you look at the basic corner filter filled with sponge used by many novice aquarium keepers, you might consider it to be just a way of performing mechanical filtration. The truth is however that the sponge is an ideal home for beneficial bacteria that can perform biological filtration. The corner filter is typically combined with a pump that causes a water flow and constantly provides the bacteria with biological waste products that are dissolved in the water. If you place a piece of activated carbon in your corner filter, it will perform chemical filtration of the water that is pumped through. This way, you can have three different types of filtration going on in the same piece of equipment.
Mechanical filtration will remove larger debris form the water. You can get internal mechanical filters as well as external ones. A mechanical filter is typically combined with a pump that forces water through some form of filter media, e.g. sponge, foam or floss. Before you purchase a mechanical filter you should always find out how much water it will circulate and how fine the filter media is. A general rule of thumb states that a mechanical filter should pass roughly twice the amount of water in the tank every hour, but this is not true for all types of aquariums.
The fineness of the filter media in a mechanical is very important. A very fine filter medium will catch even small debris from the water, but is on the other hand prone to clogging and must be cleaned very often. A coarse filter media is less prone to clogging, but will on the other hand allow a lot of debris to pass through. Many aquarium keepers use a combination of several filter media, where a coarse filter traps large particles before the water is forced through a finer medium.
During biological filtration, harmful nitrogenous compounds produced by the animals in the aquarium will be converted into less harmful nitrogenous compounds by two different types of bacteria. The metabolism of your fish will produce ammonia that is excreted through the gills and present in the fish poop. High levels of ammonia are unhealthy for fish, and really high levels can even be lethal. There is however a certain type of bacteria that feed on ammonia. This type of bacteria will ingest ammonia and excrete nitrite. Unfortunately for your fish, nitrite is even more poisonous than ammonia. For successful biological filtration, a second type of bacteria is therefore necessary. This type of bacteria ingests nitrite and excrete nitrate. Nitrate is less poisonous than ammonia and nitrite, but you must steel keep an eye on the levels of nitrate in the aquarium and remove the nitrate by performing frequent water changes.
You can support the colonies of beneficial bacteria by providing them with suitable materials to colonize, such as sponge and floss filter media, gravel, sand, plant leaves etc. When you start up a new aquarium, you should let it go through a process known a cycling. During this process, large colonies of both types of bacteria will establish themselves in the aquarium. You can jump start the cycling process in a new aquarium by introducing bacteria from an already cycled aquarium, e.g. by taking some gravel or filter media from it. You can also purchase a solution filled with the right type of bacteria from an aquarium shop. If you use materials form an established aquarium, you always risk introducing undesirable organisms together with the beneficial bacteria.
The most commonly used material for chemical filtration is activated carbon. Activated carbon will bind a wide range of different compounds to it self and thereby remove them from circulation. It is common for aquarium keepers to use chemical filtration for shorter periods only. When you have completed the treatment of a disease or parasite in the aquarium, you can for instance use chemical filtration to remove any traces of left over medication from the water. The use of activated carbon is quite controversial, and many keepers of planted aquariums refrain from using activated carbon for any longer periods of time since they suspect that it might lead to a depletion of nutrients.