Aquarium sumps are very similar to a sump on an internal combustion engine. They're basically a reservoir under the tank that holds water. The water is pumped up to the tank and overflows back to the sump, much like oil in an engine.
- Sumps are used to increase water volume in the system, which increases stability.
- Sumps are used as filters. You can create all kinds of custom filters to fit your needs with a sump.
- Sumps are used as refugiums to keep critters that don't get along with the rest of your tank.
- Sumps are used to hide equipment such as heaters from view.
I'm sure there are other uses that I'm missing. Google Melev's Reef. There's a ton of information on sumps there.