Despite the variety of filters available to aquarium owners, they all ultimately only offer one or more of the three basic kinds of filtration needed to maintain good water quality - mechanical, biological, chemical. Precisely how these types of filtration are accomplished is less important than that they exist and are effective.
Mechanical filtration is the removal of particles from the water.
Biological filtration is the conversion of fish waste products from toxic compounds to non-toxic ones. That is, the initial compound, ammonia, which is very toxic to fish in even very small amounts, is converted to nitrite, another toxic compound, and then to nitrate, which is basically harmless to fish. Chemical filtration is the removal of some types of dissolved organic compounds in the water. However, not all such compounds are removed by chemical filtration, and for this reason, among others, regular partial water changes are an important part of keeping fish.