It might be best to take a moment to first discuss some of the various hydroponic methodologies.
The simplest involves just dripping a nutrient solution over the root area of a plant which is physically supported by an inert medium like pea gravel, perlite, or the like. The nutrient is then "discarded", meaning it is not recirculated over the plant roots again. This isn't a very efficient or cost-effective method, so it is rarely used.
A wick system places plants in an inert medium suspended over a reservoir of nutrient solution. Absorbent wicks placed in the medium extend down into the solution and draw it up to the plant roots.
A water culture system does away with a growing medium and suspends the plants themselves above a nutrient solution into which their roots hang down. A popular method involves placing plant starts into small holes spaced apart on a styrofoam sheet and floating it on top of a wide, shallow reservoir. This method is often seen employed for lettuces.
A relatively recent take on water culture is the Kratky method.
Another technique is to create a bubbler system from plastic bucket-type containers. Stacking a 3.5 gallon bucket inside a 5 gallon bucket is a popular design. The smaller bucket's bottom is perforated and then placed inside the larger. The space between the bottoms of the buckets serves as a nutrient reservoir. An aquarium air pump is used to power a tube system that "bubbles up" nutrient solution from the reservoir to the top of the planting media in the top bucket where it trickles down over the roots and, ultimately, drips back into the reservoir through the perforated bottom.
A different approach employs a method know as ebb and flow. In this method, plants are placed in an inert medium in small pots which are arranged in a large six-inch deep tray. Using a timer to control a pump in a reservoir below the tray, nutrient solution floods the tray-- and the pots-- for a specific interval, and then drains back to the reservoir. An adjustable height drain determines how deep the tray is filled during the flow cycle. A grid of channels in the floor of the tray allow for complete draining. During the "ebb" time, the plant roots are kept moist by the growing medium, but are also exposed to an increased amount of oxygen.
A system similar to ebb and flow, but simpler, is called deep flow. Pots remain immersed in nutrient solution, but the solution is oxygenated and circulated constantly.
A more sophisticated method called nutrient film technique (NFT) employs the concepts of both feeding and supplying increased oxygen to the roots, but does them at the same time. Plants are situated in an inert medium in "net pots", which are plastic pots with many small slots in their sides and bottom. The pots are inserted into holes in the covers of plastic channels, known as gullies, through which a shallow film of nutrient solution flows along the bottom. The slots in the pots allow roots to grow through the pots down into the nutrient film.
An offshoot of hydroponics is aeroponics. Also a soilless growing method, aeroponics goes one step further to almost eliminate the growing medium itself. In an aeroponic system, plants are suspended with their roots hanging in an opaque chamber. Nutrient is sprayed on the roots or misted in the chamber either constantly or at specific intervals.
Even more ambitious is aquaponics, which combines aquaculture (the raising of aquatic animals like fish or crustaceans) with hydroponics. Biologically filtered water from the aquaculture system, and the animal waste by-products (nitrates) it contains, is circulated through the hydroponic system as a nutrient solution. The nitrates, which if left to build up in the aquaculture system would eventually be toxic to the animals, are removed by being taken up as nutrients by the plants. Cleaned, oxygenated water from the hydroponic system is returned to the aquaculture system. Large commercial aquaponic operations sell animals reared in the system, as well as the plant produce.
The hybrid system I built employs elements of deep flow, NFT and aeroponics. Rather than have nutrient pumped in at one end of a gully to flow past roots on its way back to the reservoir, it is instead sprayed onto the net pots at each plant site. Additionally, the drain at the reservoir end of the gully is adjustable, allowing the depth of the film flowing down the bottom of the gully to be varied from very thin all the way up to nearly filling the gully. This adjustability allows seedlings placed in the system to start out under the deep flow technique, and, as their roots grow longer, the depth can be lowered allowing more oxygen to the roots in a hybrid aeroponic/NFT system.
It is not an original design. I found a commercially available system that very apparently uses mostly commonly available materials in its construction, so I decided to build one myself. After a few hiccups, I eventually got everything I needed and assembled the system, shown here.
Another take which seeks to maximize plant density per square foot is verticulture, wherein plant sites are inserted into vertical "grow towers" inside which nutrient solution is dripped over roots and grow media.