A sump is basically a second tank that is connected to your main tank. Water is moved from the main tank to the sump, and pumped back up to the main tank. The sump is used to hold all your equipment (skimmer, heater, etc) so that the equipment doesn't have to hang off the back of the main tank. This makes your tank look much nicer!
Like Mo said, a sump also increases the total volume of your tank. So if you have a 55 gallon tank with a 20 gallon sump, your total system volume is 75 gallons. This means that waste from your fish will be diluted out more. So having a sump increases your water quality.
Like Mo also said, you can use a section of the sump to keep macroalgae, which take up nutrients from the water that would otherwise fuel nuisance algae blooms in your main tank.