What is an air pump?
Air pumps are a great way to increase the dissolved oxygen in your aquarium water. They sit outside the tank (in the cabinet underneath or tucked behind it out of sight) and push air through a narrow hose into the water. They can connect to an air stone (a porous block that turns big bubbles into tiny fizzy bubbles) or into an ornament - you can get some really cool ones with parts that move as the bubbles flow through them.
Are they essential?
No, if you have a good filter and your tank isn't over-stocked. They are very useful for adding extra oxygen when you need it, e.g. if your fish are poorly, extra oxygen in the water helps them breathe without having to waste energy on swimming. It is also important to increase the oxygen level if you are ever adding medication, as the medicine dissolved in the water means there isn't as much dissolved oxygen available. Many fish enjoy 'playing' in the bubble stream, so it can be a useful enrichment tool, and obviously if you like the bubble-effect then that's all the argument you need!
What else do you need with them?
The pump feeds air into the tank with 'airline' hose, so you'll need a length of that. You also need a 'check valve' or non-return valve' which is a one-way valve to stop water back-siphoning along the tube and onto your carpet if you have a power cut and the air flow stops. You can get all sorts of connectors to split and join air tubes, and little taps to control the flow. Airstones come in different shapes and sizes depending on the effect you want.