Aquaponics is a combination of hydroponics and aquaculture retaining the benefits of both while excluding the negative aspects.
Basicly aquaponics is growing fish and plants together in unison and harmony a cycle, fish waste feeds the plants, plants clean water for the fish, everyone's happy.
The three most commonly used systems are media filled beds, Nutrient Film Technique (NFT) and Deep Water Culture (DWC).
Media filled growbeds are the most commonly used in household systems where containers are filled with gravel or expanded clay. Water from the fish tank is pumped into the growbeds and the plants grow in the growbed medium. There are many variations to this system there are continuous flow, flood and drain or the use of an auto siphon.
NFT involes no grow medium but rather a thin 'film' of nutruient rich water in enclosed containers with the roots of the plants hanging in the water. This works rather well and is quite cost effective but is only suited to small leafy vegetables such as lettue due to enlarged root devlopment blocking the waterways will lead to problems later on.
Deep Water Culture works by floating plants above a body of water with only the roots in the water. This method is commonly used in large commercal systems but can be as simple as a floating basket of mint in some water.