Okay professer,
Generally a refugium can be looked at as a separate area of the saltwater tank system, that is isolated from the main display tank.
This area allows plants and living creatures to grow and reproduce without the risk of being eaten or destroyed by the main inhabitants of the display tank.
A true refugium normally implies a viewable area such as a smaller tank, that's connected into the main tank water system much like a sump is, that you would keep these plants and creatures.
In this hobby it's also understood to be a place to grow macro algae such as caulerpa, a sandbed to grow pods and worms and such, maybe contain some live rock, or a combination of all of the above.
A deep living sandbed in the refugium may also provide a place for anaerobic bacteria to consume excess nitrate, otherwise known as "denitrification". Same concept as having a deep sandbed in the display tank - but some choose to just do this in the refugium. Naturally some folks have a deep sand bed (DSB) in both the main display tank and the refugium.
Lower nitrates leads to better overall water quality and less chance for nuisance algae in the main tank.
People set up a refugium for lots of different reasons, but I would say the most popular reasons are for nitrate reduction as mentioned, and to grow pods and other small critters.
As the fish/inverts are fed and excrete wastes in the main tank, eventually in most every case, the tank will have a rise in nitrates. In addition to having the DSB and live rock in the refugium, the macro algae or caulerpa placed in the refugium will use this nitrate ( and phosphate ) as it grows. That's what most living plants do. They use this stuff as "fertilizer", and along with having light, water and carbon dioxide .... remove the nutrients by taking them up in their growing plant tissue, and give off oxygen gas as a product of photosynthesis.
Lowering the carbon dioxide or CO2 in the tank water can also help to prevent the pH of the tankwater from dropping so much over time, and helps to stabalize other chemical reactions that typically occur in everyone's tanks.
Adding O2 oxygen is good for the fish, inverts and bacteria as I'm sure you know.
As they grow - the hobbyist "harvests" or removes some of this algae now and then, which "removes" the nitrates and phosphates from the water system.
It's what many refer to as "nutrient export."
Just think of a refugium as a little tank or container, that shares the water with the main tank, but does not allow any fish or other predators a chance enter it. It's a protected zone, where you can grow the "good" algae, and the copepods, amphipods, worms and other sandbed critters can grow with out getting munched up by the fish. Some of these little critters make their way to the main display tank, and help to sustain the population in it's sandbed.
Of course some of them are not so lucky, and get eaten by the fish anyways - which ain't so bad if your the fish
If you really are serious about setting up a sump and refugium, we would need more information regarding your current set up, size of tank, type of existing filtration, free area under the tank stand, whether you want to "purchase" everything, or if you plan to DIY do it yourself.
Please provide as much info on your tank and current set up as you can, and if you are comfortable doing some of the work yourself.
From there we can start laying out a plan.