A closed loop is a way to circulate a large amount of water for the purpose of generating current for the corals in our tanks. They are most commonly used on sps tanks where these corals require alot of flow and prestine water conditions, however any tank will benefit from adequate water movement that is constructed to match the requirements needed by the tank's inhabitants.
They also allow for division of flow generation between the closed loop and the sump. Essentially, you can slow down the flow throught your sump to allow more time for a skimmer, refugium, carbon media, etc to do their job by creating more contact time with the water in the sump, as opposed to continually circulating it to provide adequate flow for the corals in the tank.
Closed loop also means that the plumbing of system is "closed" to air, unlike a sump which is open to air or an "open" system.
Hope this helps.