A sump is just an additional reservoir of water. This reservoir is plumbed inline with your main tank, and is normally (if not always) located under your main tank. Water from your main tank flows out of the main, down to the sump, and then is pumped back up into your main. That's pretty much it in a nutshell. There are numerous ways of getting the water down to, and back out of the sump - which is normally where it gets confusing.
A sump gives you two things:
1. An additional reservoir of water for a given tank size. Say you have a 75g main tank with a 30g sump. Even though your main is only 75g, you have the water capacity of a 105g. This gives you more stable water parameters as there is more water to dilute any of the bad stuff. (On the flip side, if you add additional calcium or any trace elements to your tank, you'll have to dose more because you have that additional water in your system.
2. A place to hide heaters, skimmers, calcium/phosphate reactors, etc.