A sump is merely an additional reservoir for water that's plumbed inline with your main tank. It increases the capacity of your system, and also allows you to stick equipment like heaters and skimmers in that area to get the clutter out of your display tank. Sumps are obviously not required, but going along with "dilution is the solution to pollution", more capacity is always better.
A skimmer is a man made way of recreating ocean surf. It basically makes a bunch of bubbles and forces the bubbles up a tube into a collection chamber. Organic compounds in your water adhere to the bubbles, and are pushed up and out of the water into the collection cup. These organic compounds, if left in the water, will eventually degrade your water quality. So in a nutshell, a skimmer takes stuff out of the water before it can decompose and make your water quality bad. In smaller tanks (say less than 20g) skimmers are often not used and more frequent water changes are done to insure pristine water.