A sump is another area for your tanks water to drain to, and then be returned with a pump. It adds to the overall water capacity of the system. In other words, a 30 gallon sump underneath a 75 gallon makes for a 105 gallon system. You can also run a skimmer, heater, or whatever other equipment will fit in the sump, so you don't have to have it all hanging on your tank. Makes it look much nicer, and makes it a lot easier to make a nicely sealed lid. This would also be an ideal place to run a fan to cool the water, since you aren't going to get a very good breeze over a tank with a sealed lid. A lot of people also section off a part of the sump to use as a refugium. A refugium is ideally a "refuge" for things that would be eaten in the display, like copepods (basically tiny saltwater bugs) and in addition keep a sandbed, some liverock, and macroalgae. The point of that is the macroalgae absorbs the nutrients out of the water that would otherwise build up and become toxic to fish, or in this case, an octopus. Of course there is no way to completely stop that, which is the point of water changes, and even still, the macroalgae needs to be pruned occasionally, like weeds in a flower bed. Otherwise it will eventually die and release all those nutrients it has absorbed back into the water.
Here is a really good link for further explanation:
http://www.melevsreef.com/what_sump.html