WT808
It will help aerate and cycle the water and during hot summers it can help cool down the water. There is some that uses battery(2 to 4 D batteries) and starts up right when there is a power failure but still needs a outlet to detect it. It can also decorate the tank with bubbles like for example you can put one long row of a Air Stone Bar in the back of the tank or on both sides and have a nice background or put a long row in the middle and have a bubble wall and etc. WT

Ash
It can depend on what type of filtration you use. Sponge filters require air pumps to run. If you have powerheads you probably would not need a pump but it is up to you if you want to use one additionally for aesthetic purposes and for extra aeration for the fish. As long as the surface of the water is all in constant motion you shouldn't particularly need to use one, however some fish require lighter currents such as bettas and do fine with a small trickle of bubbles from an air stone rather than the ferocious current of a powerhead.

Peter P
Yes they provide more oxygen and help remove bad gases from the water. Not only that, but they make the tank look nice too.

aquaria_care
Unfortunately, if it is not driving some sort of filter, the benefit of an air pump is mostly decorative. It doesn't add much oxygen to the water. A power filter does that much better, especially if you don't over fill your tank. Fill only to the very edge of where the water cascades from the filter, you will be introducing plant of oxygen. In my own tank, I have two air stones powering an undergravel filter. For some fish, mainly giant danios, these bubbles are very fascinating to the fish. They just stop and fixate on them for long periods of time. I imagine for them, this is their deity.

Ricky H
It provides oxygen for your fish making it easier on their gills. You don't really need one if you have a filter, but it would be better for you fish if you had a filter as well as an air pump. You could hook it up to an air stone or bubble wand.