All fish require oxygen to be saturated in the water in order for them to survive. This does vary by species and is particularly critical at higher water temperatures. More often than not, a water fall return is not adequate oxygenation.
Air pumps add oxygen which is important for some species (vital for Sturgeon and Trout) to thrive as the water gets warmer. Oxygen saturation decreases with water temperature and an air pump ensures that the maximum saturation is achieved.
Air is delivered via a hose to an air stone or disc to create a bubble effect in the pond supporting the oxygen the fish like.