Aquaponics is the combining of aquaculture (growing fish in tanks) and hydroponics (growing plants in water.)
The most simple system consists of a fish tank with a "raft" floating on it into which plants are grown. The roots of the plants grow into the water and get their nourishment from the fish wastes. The only input is food for the fish. More elaborate systems will add separate grow beds for the plants.
There are two basic types of aquaponic systems, raft or flood and drain. Most commercial systems are of the raft type. These can be quite complicated, with fish tanks, clarifiers, mineralization tanks, sumps and rafts. These allow for the highest number of fish and grow the greatest number of plants. They are also the most expensive to set up.
A flood and drain system works by pumping water from the fish tank to grow beds filled with gravel or some other inert material to flood the bed. Then the pump shuts off while the bed drains. This is repeated every hour or so.
We have built a small flood and drain system to gain experience. Now we are building a larger raft system that will consist of a 700 gallon tank and 192 square feet of grow bed.