Aquaponics, simply, is the symbiotic cooperation of aquaculture and hydroponics. Traditionally in aquaculture, fish raised in holding tanks will require a routine water change as ammonia slowly builds and becomes toxic. Disposal of this water becomes necessary and if not handled in a responsible manner, can pollute the local environment.
Hydroponics is the art of raising plants in water, sometimes using a grow media like expanded clay pebbles for the roots to grasp. By providing (usually) commercial fertilizers to the water reservoir, plant growth can, in the right environment, exceed growing expectations.
Combining both aquaculture and hydroponics creates a (loosely) closed loop system where as the fish provide the nourishment for the plants by the nitrogen cycle and in return, plants clean and filter out the ammonia sending clean water back to the fish.