Most aquaponic systems are single loop and fully recirculating. This means that the fish and plants exist in a single loop and share the same nutrient-rich water. The reason these are the most popular is because they are simple to build and operate. But single loop systems are also a compromise because the plants, fish and bacteria must all be happy in the same solution. So for example, the temperature and pH levels must be maintained at levels that satisfy all the organisms, rather than being optimized for each.
In a dual loop system, these parameters can be maintained optimally at different levels in each loop. Additionally, if something goes wrong, there are more options to address the situation without affecting everything in the system. So for example, if pests need to be treated on the plant side, it can be done without adversely affecting the fish.
A typical decoupled system consists of a stand-alone RAS (recirculating aquaculture system) loop and a hydroponic plant loop. The sludge from the RAS system is digested in the biological waste system which provides the nutrients for the hydroponic system. Using a digester makes the system more efficient because the nutrients plants need (aside from Nitrogen) are trapped in the solid waste. The aerobic digester (a fancy term for a simple tank with air-stones in it) mineralizes the solid waste, releasing the nutrients the plants need.