Aquaponics is a combination of aquaculture (the growing of fish) and hydroponics (the growing of plants in water without soil). This integrated system works so well because the two form a symbiotic relationship; and nutrients and resources are recycled between the two. The fish provide waste, which the bacteria turn into plant nutrients, providing the plants with food and the fish with clean water. Indeed, bacteria is the linchpin which keeps the whole system running.
With regards to the set-up, it is actually very simple. You have the fish tanks, in which you grow fish, and on top of the tanks sits the grow bed. The grow bed is the surface in which the plants are rooted and is the location of most of the bacteria in the system. The grow bed can be made of different materials, but a common one is gravel. Gravel is ideal because it filters the water and provides support to the roots. In addition, it has a large surface area and can house a large number of bacteria.
In an Aquaponics system, water is pumped from the fish tank to the grow bed, where the bacteria reside. The bacteria break down the fish waste and convert it to nitrates, an excellent fertilizer. The plants take up the nitrates and some of the water, and in the process filter the water. The water is then returned cleaned to the fish tank via gravity.