In dealing with under-tank sump filters, you basically have a choice between two types, or some combination of the two.
The first and simplest type is simply a box (or tank) that often contains a filter sock at the output end of a gravity fed overflow system from the aquarium above to trap large particles, then mechanical filter medium, usually in progressively smaller grades and layered vertically, that catches finer and finer particles from the water as it passes horizontally through to the last area that holds a pump, which returns the filtered water to the aquarium above. Other stages of filtration, such as chemical (as with activated carbon) and biological (as with "bio-balls" or other platform for bacterial growth) are often added after the mechanical filtration. The final area often contains other components, such as a heater, water parameter sensors or other systems that further condition the filtered water (such as adding CO2) before it's returned to the aquarium. A shielded in-line ultraviolet light unit is sometimes inserted between the return pump and the aquarium to further "sterilize" or "clarify" the water to one degree or another.
The second type of sump filtration is the "wet/dry" system, the purpose of which is to saturate the water with as much oxygen as possible so that the growth and activity of nitrifying bacteria is greatly enhanced. The various methods for doing this involve systems where water is passed through stages of filtration similar to those mentioned above, with the difference being that the water is dispersed over the filter medium in droplets or thin sheets, which creates vastly more water surface area, allowing for better gas exchange between the water and air. (This natural gas exchange normally increases the oxygen level and decreases the carbon dioxide level of the water.) The methods of doing this include towers, where the water is dripped through horizontal layers of filter medium (with air spaces between the layers) so that the medium remains wet, but not submerged (hence "wet/dry"), or some other means of achieving the same goal.
A sort of "hybrid" of these two systems is achieved by creating chambers of the different filter mediums separated by baffles. The water flows over and under these baffles and through the filter medium, causing turbulence at the surface between the chambers, which results in enhanced gas exchange. While the oxygenation effect and resulting "supercharging" of the bacterial colony is not as great as with the true "wet/dry" system, increased oxygenation and its benefits still occur to some degree.
The wet/dry system is often used with marine aquariums, where a constant high oxygen environment and intense biological filtration are desired to maintain the highest levels of water quality demanded by many ocean-dwelling species. This type of system may also be desirable for a freshwater aquarium in which the focus is on the fauna, and most or all of the flora is strictly decorative and artificial.
In a freshwater planted aquarium, the focus may be evenly divided between flora and fauna, or even skewed more or mainly toward the plants. In this case, a high-oxygen, low-carbon dioxide environment is not preferable, due to the plants' need for carbon dioxide, so the first type of filter is usually chosen because it does not off-gas carbon dioxide through accelerated gas exchange before the water is returned to the aquarium, plus the fact that the breakdown or removal of waste products from the water that is usually thought of as the domain of the beneficial bacteria is, in part, assisted by the plants (and, in fact, the plants benefit from doing this), so the extreme activity of the larger beneficial bacterial colony is not needed.
I think that boils it down to the basics of sump filtration, although having a part of the aquarium environment located out of sight and in a space of it's own also allows for many other options.
"May the Fish be with you."