It is obvious to most people that aquarium filters capture the gunk suspended in the tank’s water, leaving it looking clear. This is called mechanical filtration. Filters perform a much more important function, however, called biological filtration. As discussed in the post on cycling an aquarium, fish produce ammonia both directly and through the decomposition of their waste. Ammonia is toxic, but the cycling process builds up a colony of bacteria that consume the ammonia to produce nitrite. The nitrite is even more toxic than ammonia, but is also consumed by bacteria to produce nitrate, which is much less harmful.
Filters perform the important task of moving water that contains ammonia and nitrite past the colonies of bacteria. Insufficient movement of water can result in a toxic build up of ammonia or nitrite, which can’t be processed quickly enough. This mostly happens in tanks that are overstocked with fish or over fed with too little filtration and water movement.
[...] the aquarium has the proper amount of filtration. While there are a number of variables, such as volume of water, number and types of fish, [...]