The basis of aquaponics is the combination of aquaculture and hydroponics. For those who may not be familliar:
Aquaculture is the raising of fish on an industrial mindset for the purpose of food. The fish can be raised in higher densities than would be found in the wild, and there is no chance of failing to harvest as there would be with fishing for food. The proteins gained from the fish meat can fill a void in a person's diet.
The drawbacks: Often requiring large water sources, the process can be damaging, as the toxic ammonia produced by the fish is, in many cases, simply carried away by natural waterways. If natural flows are not used, the systems often prove wasteful, as the waste must constantly be filtered out and disposed of by other methods. In addition to the feeding of the fish, this practice can often prove ineffective for survival situations.
Hydroponics is the practice of growing plants in a soil-less medium: Water is pumped past the roots of plants that are either suspended in the water or planted in some medium, such as gravel. The advantages are rather obvious: The plants can benefit from incredibly accelerated growth rates, and can grow as such in relatively high densities.
However, in most cases, the water must be heavily laced with fertilizer. Not only is this damaging to the surrounding environment when the water must be replaced, but constantly supplying fertilizer can be expensive, inefficient, and in a survival situation, literally impossible.
Aquaponics is when the two above-mentioned practices are combined. Water from an aquaculture set-up is pumped to a gravel bed, where any variety or assortment of plants can take root. The photo-phobic bacteria in the gravel bed, Nitrosomonas, converts the toxic ammonia, NH3, into nitrites, NO2. This is then converted from nitrites into nitrates, NO3, by another bacteria known as Nitrobacter.
Ammonia (NH3) --> Nitrites (NO2) --> Nitrates (NO3)
The nitrates act as a direct, and incredibly powerful, fertilizer for the plants, with some growers claiming up to 18 times faster growth rates. Therefore, this system eliminates the need for adding chemical fertilizers to support the plants, as well as the changing of the water to avoid detrimental chemical buildups.
In addition, as the plants are, in essence, absorbing the dangerous ammonia, the need to exchange or filter the water that holds the fish is removed as well. On top of this, a separate tank can be added to the flow of water, allowing the growth of plants such as Water Lettuce and Duckweed; feeding the fish these, along with Black Soldier Fly larvae, will supply all necessary nutrients, and can also be fed to other livestock, such as chickens.

I built an aquaponics system to see firsthand if the fantastic claims of aquaponics held any truth. I used a simplified version of the popular Barrelponics to create the environment which would house my fish and plants. We have many 55 gallon HDPE barrels that had once been used for water storage. Using two barrels, recycled landscaping gravel, unused garden hose, scraps of 2x4, an old pond pump, and left-over cinder blocks from building a garage, the system itself literally cost me nothing to build. In the end, I paid roughly five dollars for seeds, and around twenty for three fancy goldfish.
This is the initial set up. Before gravel, fish, plants, etc are added. The growbeds are composed of the two halves. The pump is submersed in the vertical, topless barrel, known as the reservoir. The beds fill continuously, until the water reaches the top of the hose; a suction is created, and the barrels drain completely. Once empty, the suction is broken, and the pump continues to refill them without pause. This cycle of water and air provides nutrients, water, and air to seeds that can be directly planted into the gravel. 
This photo was taken roughly four weeks after planting. The lettuce, which was predicted by the company to sprout within a week and a half, sprouted within two days. The beans took roughly a week, instead of a week and a half, and instantly grew four to six inches, before the growth rate slowed and steadied. The cucumbers, hidden here behind the lettuce, took the two weeks that was recommended of it, before growing to the size of a small soccer ball, and then ceasing to grow. It took another two weeks, before the cucumbers suddenly flowered, and then exploded in growth again - the inconsistency is undoubtedly due to ammonia levels fluctuating in the water. Directly in front of the beans, several pepper seeds were planted. For reasons unknown, they never sprouted; however, I am of the impression that this is due to factory defect as opposed to unsuitability in the system. 