A sump, when linked to a fish tank, is essentially just a secondary tank positioned somewhere below the main tank and that's fed with water through gravity. The water is returned towards the main tank using a pump once it has been processed in the sump. Generally, the level of the key tank will pass through the sump a couple of times an hour or so. The sump itself could be configured in several new ways to provide specific functions that benefit the main tank in some manner.
First of all a sump, during it's simplest form, adds volume to the system. If the main tank is 100 gallons and also you give a 50 gallon sump, well in that case the volume of the entire system rises to 150 gallons. With this added volume comes added stability. A bigger level of water takes longer to improve in temperature, salinity, or whatever parameter you want to use. And as I've said again and again, stability is essential with a healthy aquarium.
After adding volume, the subsequent most common reason to include a sump inside your aquarium refugiums is always to offer you a destination to place all the apparatus that runs the thing. Filters, heaters, skimmers- it could all use the sump. This implies less clutter within the tank or hanging off the back than it. Even more so that it could be the only option if the back with the tank fills up and you still have equipment which needs to be connected. Furthermore, considering that the sump is probably located in the enclosed stand the noise everything that equipment generates will be reduced also.
All sumps are fed by some type of overflow mechanism either hanging around the back of or constructed into the tank. This mechanism is made so regarding allow water in the tank spill over into it if this gets too high and flow right down to the sump. The benefit of this really is that the top of the water in the tank is consistently skimmed clean. Tanks without an overflow frequently have a greasy film of proteins and oils floating at first glance of the water that is problematic as it can block gas exchange. By having an overflow, this layer is pulled into the sump and churned into the water for the protein skimmer to deal with. Additionally, that churning also helps increase gas exchange - enhancing the dissolved oxygen level of water.
A sump entails a far more stable level in the primary tank. Marine aquariums specifically lose a lot of water to evaporation. On setups with out a sump the water level inside the tank drops as water evaporates, possibly exposing intakes or other equipment within the tank (or even corals which have grown very tall) for the air. Not to mention even though everything is low enough never to be affected you'll still end up seeing the reduced level externally frequently which, without exactly a tragedy, isn't pretty either.
Perhaps the best benefit of a sump that isn't immediately recognizable would it be gives you a good place to introduce additives towards the tank. Reef tanks typically need daily doses of calcium, alkalinity, and/or other supplements to maintain the water's parameters under control. Several chemicals are highly concentrated and if added straight to the tank need to be added very slowly. Using a sump to just dump them in to be diluted down before they go into the tank makes adding them much less of the headache. Likewise topping off evaporation is a lot easier with a sump for the similar reason. Relatedly, a sump makes a good way for your heater and/or chiller since the localized hot/cold spots they produce is going to be safely out of the inhabitants of the tank.