Porter is actually the great-grandpa of today's stout. It was, and still is, made with dark malted barley, a good amount of hops, and top-fermenting ale yeasts. The end result is usually a dark medium-bodied beer with a nice balance of malty sweetness and bitter hoppiness.
Stouts came about when people started tinkering with their porter recipes. They started adding new ingredients and upped the alcoholic strength. At first these were called "stout porters," but eventually the "porter" was dropped and stouts became a category unto themselves.
Today, the biggest difference between stouts and porters is the kind of malt used during brewing. While porters use malted barley, stouts primarily use unmalted roasted barley. It's this ingredient that gives stouts their signature coffee-like flavor. Porters also tend to be slightly lighter and less full-bodied than stouts.
Having said all this, there is definitely a lot of cross-over between porters and stouts. If you like one, you'll generally like the other! The best is to grab a friend and get one of each to compare and contrast.
What do you like about porters and stouts? Any recommendations for ones to try?
(P.S. The picture above is a porter. Kudos if you guessed it correctly!)