There are many different types of systems. The flood and drain system described by Earthling is one, very popular, style of system. It success is due to the fact that the grow bed where the plants are living is slowly filled, and then quickly drained. This quick draining pulls oxygen into the bed. This effectively is providing three things to the plant in a constant fashion:
- An unlimited supply of oxygen
- Nutrients from the fish (more on how in a second)
- An unlimited supply of water
The nutrients that the plants need come somewhat indirectly from the fish. The fish go the bathroom. This causes the ammonia levels to build up in the water. There is a bacteria that will establish itself in the system which will convert this ammonia into nitrites. Once the water starts to have a certain level of nitrites in it another bacteria will establish itself. This second set of bacteria convert the nitrites into nitrates. The nitrates are what feeds the plants. This bacteria lives in the edges of the substrate/media. This media is usually a type of rock or manufactured rock.
But this is the first half of the story. The process described above feeds the plants. But it is also helping to clean the water for the fish!
The water moving around splashing all over in the system is what cleans out the carbon dioxide that the fish deposit in the water as they breath. Sometimes there are issues in having enough of this process present in the natural motion of the system in which case a bubbler is also introduced.
The only input to a well crafted aquaponics system is the food for the fish. But clever people have figured out ways around that too using black soldier fly larvae!
Other systems you might look at, which are similar in process, are grow towers. In the towers case you would use a sponge like material similar to what you would find in a bio filter in an aquarium. The ideas being the same.
Then there are systems that don't use grow media at all. NFT and DWC are a couple of examples. These are closer to hydroponics. The water is constantly on the roots of the plants without ever having a break. In this type of system you need two additional concepts. You need a media based bio filter similar to that of the towers but you wouldn't plant your plants in them. The water flows through them allowing the bacteria to clean the water. And immediately after the fish tanks you have some form of solids filter. In many cases this comes in the form of a swirl filter where the water goes round and round allowing the big poops to fall to the bottom of the tank. In a system like this you usually would have to remove the big solid waste now and then. Some folks dump this directly into a grow bed so as not to have a waste being generated.
NFT: Nutrient Film Technique - the plants sit in a very long and very shallow tube with holes cut in it or tray (much like a rain gutter) that has a top on it also with holes cut in it. Nutrient rich water is trickled from one end of the tube to the other end in a constant flow - just enough water for the roots to drink from. Plants sit in the holes every foot or so (depending on what you are growing) with their roots just touching the bottom of the tube. This allows maximum fresh air to get to the roots as the water pulls it in as it travels. The roots eventually grow into the tube and lay in the water. Two immediate down sides to this technique is that 1) without appropriate filtering, solids can get stuck on the roots and eventually cake the root disallowing uptake of nutrients, water, and oxygen and 2) in hot and cold climates the water that enters the tube is at the temperature you specify but can be drastically different once it reaches the other end some 50-100' away. This can cook the roots or freeze out the roots.
DWC: Deep Water Culter - this is very much the same thing as NFT in principle but on a totally different scale. Two big differences with this approach is that rather than planting in a small tube you plant in a huge trough of water. The trough might be equally long 50-100' (or more) but the trough is also 4-5' wide and at least 12" deep. The water also enters one end and exits on the other end. Another major difference is that the plants sit on a foam (usually) raft with holes cut in the raft. The plants sit in plastic cups and almost their entire root body hands into the water totally submerged. Filtering is still an issue but in aquaponics there are many that are running their water through a grow bed or grow tower prior to entering the DWC tank. This way you don't loose the good stuff that is filtered. And because of the sheer volume of water it doesn't tend to fluctuate in temperature nearly as much. Also - having that much water means that keeping the chemical balance is usually easier. One down side in some dwc systems is lack of oxygenation. You can design bubbling spots in your plumbing and system but many people add additional pumps with bubble stones directly to the grow tank which aids in off gassing.