Fish, like us, need oxygen to survive. Most fish derive ogyxen from water through their gills. Some also have the ability to air-breath.

Oxygenation of the water is irrelevant to most marsh-dwelling fish. Bettas and paradise fish can survive in completely stagnant water with hardly any dissolved oxygen in it. Most apistogramma or boraras species come from similar biotopes. So do many gouramis. Corydoras, too, gulp air.

However, fish from rivers and lakes need oxygen-rich water and will perish without. This true for the African Rift Lake cichlids like Pundamilia nyererei. Also think of danios, loaches, etc. A common European river perch will not survive long in a bucket for this reason.

For most fish tanks, a hang-on-back or external filter is a good idea - apart from its obvious filtration merits, I mean. If its outlet is in or above the water surface, air will visibly enter the water column, leading to oxygen dissolving in it. But an underwater outlet will have much the same effect, as it will cause surface agitation, which leads to oxygen absorption. A layer of biofilm on the water prevents effective oxygenation of the water and should be avoided in tanks with riverine or lake species.

An air-powered filter creates oxygenation and surface agitation, but only little current in the tank, which makes it more suitable for breeding and grow-out tanks, as fry may otherwise be sucked into the wheels of the filtration system.