A number of filtration methods exist.
Under-gravel filtration used to be the most popular and is very effective. This type of filtration draws oxygenated water through the gravel which sustains a colony of bacteria. The bacteria process waste products in the water. The water can be circulated using either a water pump or a lift tube and air pump. This method is very effective for tanks where water currents may cause a problem, such as a tank used to raise fry.
Under-gravel filtration has been mostly replaced by various types of power filter. These contain various permutations of sponge (or filter wool), filter carbon and pellets.
The sponge type media removes particulate debris from the water, and when used correctly can maintain a bacteria colony. Filter carbon (or “activated carbon”) absorbs various chemical traces from the water, and must be removed when any type of medication is added to the water. Pellets provide a media for a bacteria colony to occupy, which process various wastes – some bacteria species process ammonia (a fish waste) to produce nitrite, another species process nitrite to produce nitrate. Nitrate is the least toxic to fish, and is typically removed by plants to use as food, or can be removed by partial water changes.
Power filters are generally easier to clean than under-gravel filters, but have the disadvantage of moving parts. Care must be taken when configuring the outlet of power filters, which often have high flow rates that are troublesome for some species of fish.
Your aquarist will be able to help you select a filter appropriate for your tank size. He/she will also probably recommend a power filter, since these are generally the most effective.
If using a filter that uses bacterial action (which will almost certainly be the case), bacteria will need to be introduced to the tank before the fish. Traditionally, a small amount of food would be placed in the tank with no fish, which would decay to “kick-start” the filter. These bacteria can now be purchased in bottles, and I recommend this approach.
Since a sponge may also house bacteria, it should be cleaned in a way that will not destroy the colony. Only clean one sponge in any given maintenance period. If the filter only has one sponge, consider cutting it in half with a pair of scissors. This will make sure bacteria will always be present. Clean the sponge by rinsing it it aquarium temperature water (this is most easily done using the waste water during a partial water change).
Only use a sponge provided by your aquarist dealer. Some types of domestic sponge contain chemicals that may be harmful to fish, others may release micro glass fibers into the water.
Note that filter output can be used to keep water circulating past a heat – keeping the temperature even across the tank.
Whichever solution is adopted, a UV water steriliser needs consideration. These compliment (not replace) filtration. Once only common in garden ponds, small units for the aquarium are becoming inexpensive, and resemble a power filter. These kill any micro-algae suspended in the water (keeping the water clear), and arrest the spread of disease. I consider my UV steriliser indispensable.