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There are two major types of filtration for your fresh and saltwater aquarium. Mechanical and Biological. Mechanical filtration is the foam and wool that physically removes debris, food and waste from the tank. Biological filtration is the media and absorbants that absorb good and bad bacteria. The Biological filter takes the ammonia produced by fish waste and rotting food and turns it into Nitrite, the Nitrite is then turned into Nitrate by the biological filter. Both Ammonia and Nitrite are toxic to fish, however Nitrate is non toxic. High levels of Nitrate can stress fish and stressed fish will stop eating and die. Which is the main reasons for water changes.

When a filter is operating the way it should, the desired result is as follows. Fish produce ammonia, the biological filter turns the ammonia into toxic nitrite then the biological filter turns the nitrite into non toxic nitrate. The levels of nitrate in the aquarium slowly rise and then you do a water change to remove the unwanted nitrate.

when a filter is not coping the ammonia, nitrite and nitrate levels will rise in a tank. You can de-toxify the ammonia and nitrite and do more frequent water changes until the biological filtration balance is restored.

Wikipedia has the following to say about filtration.

Overview
Animals, typically fish, kept in fish tanks produce waste from excrement and respiration. Another source of waste is uneaten food or plants and fish which have died. These waste products collect in the tanks and contaminate the water. As the degree of contamination rises, the risk to the health of the aquaria increases and removal of the contamination becomes critical. Filtration is a common method used for maintenance of healthy aquaria.

Biological
Proper management of the nitrogen cycle is a vital element of a successful aquarium. Excretia and other decomposing organic matter produce ammonia which is highly toxic to fish. Bacterial processes oxidize this ammonia into the slightly less toxic nitrites, and these are in turn oxidized to form the much less toxic nitrates. In the natural environment these nitrates are subsequently taken up by plants as fertilizer and this does indeed happen to some extent in an aquarium planted with real plants.
An aquarium is, however, an imperfect microcosm of the natural world. Aquariums are usually much more densely stocked with fish than the natural environment. This increases the amount of ammonia produced in the relatively small volume of the aquarium. The bacteria responsible for breaking down the ammonia colonize the surface of any objects inside the aquarium. In most case, a biological filter is nothing more than a chemically inert porous sponge, which provides a greatly enlarged surface area on which these bacteria can develop. These bacterial colonies take several weeks to form, during which time the aquarium is vulnerable to a condition commonly known as "new tank syndrome" if stocked with fish too quickly. Some systems incorporate bacteria capable of converting nitrates into nitrogen gas.[5]
Accumulation of toxic ammonia from decomposing wastes is the largest cause of fish mortality in new, poorly maintained, or overloaded aquariums.[6] In the artificial environment of the aquarium, the nitrogen cycle effectively ends with the production of nitrates. In order that the nitrate level does not build up to a harmful level regular partial water changes are required to remove the nitrates and introduce new, uncontaminated water.

Mechanical
The process of mechanical filtration removes particulate material from the water column. This particulate matter may include uneaten food, faeces or plant or algal debris. Mechanical filtration is typically achieved by passing water through materials which act as a sieve, physically trapping the particulate matter.[1] Removal of solid waste can be as simple as physical hand netting of debris, and/or involve highly complex equipment. All removal of solid wastes involve filtering water through some form of mesh in a process known as mechanical filtration. The solid wastes are first collected, and then must be physically removed from the aquarium system. Mechanical filtration is ultimately ineffective if the solid wastes are not removed from the filter, and are allowed to decay and dissolve in the water.
Dissolved wastes are more difficult to remove from the water. Several techniques, collectively known as chemical filtration, are used for the removal of dissolved wastes, the most popular being the use of activated carbon and foam fractionation. To a certain extent, healthy plants extract dissolved chemical wastes from water when they grow, so plants can serve a role in the containment of dissolved wastes.
A final and less common situation requiring filtration involves the desire to sterilize water born pathogens. This sterilization is accomplished by passing aquarium water through filtration devices which expose the water to high intensity ultraviolet light and/or exposing the water to dissolved ozone gas.

Hope that helps.