A refugium is separate tank, or refuge, attached to the main aquarium system. The purpose of a refugium is to serve as a breeding ground for algae and tiny invertebrates. They are able to grow faster in a refugium because there are no predators to eat the algae or inverts. These tiny inverts such as copepods can eventually make their way into the main display tank where they will provide food for corals and fish. Mandarin fish feed almost exclusively on copepods and will thrive in aquariums with refugiums. 

Why would you want to grow algae in a refugium? If you grow algae in a refugium, it uses up the nutrients it takes for algae to grow in your main display. The result is less unwanted algae in your display. Also, algae and other plants feed on the nitrates in your water. The result is that your tank will require less water changes to get rid of nitrates. Livestock can live and thrive in water with nitrates less than 30 ppm, but too much nitrate is harmful. To grow algae, your refugium will require lighting. Many hobbyists run a reverse light cycle (opposite of the lights on the main tank) on their refugium, or even keep the lights on 24 hours a day.  

Also, some people set up refugiums to keep specimens, such as mantis shrimp, that they do not want in their display tank. Refugiums can be a fun way to keep odd inverts and animals that may create havoc in a reef tank.