If you'd like to read an interesting, fairly in-depth explanation of the differences between the two products (including side-by-side baking tests and information about crumb and flavor), then I suggest that you hop over to the Huffington Post.
(Photo By: Kate Ter Haar; Baking Soda reacting to an acid)
If you just want to have a basic understanding, here you go:
Baking Soda is a single ingredient baking base that causes lift and browning in your baked goods when combined with an acid. The reaction is immediate so you should only combine your acid to the base when you're ready to put the baked good into the oven. It does not expire.
Baking Powder is a three ingredient baking item that contains both the base and the acid. This allows you to add lift to you baked goods without having to balance acid to base ingredients in your recipe. Additionally, baking powder usually is "double-acting" which means there will be initial lift when the baking powder comes into contact with liquid and then another lift with the batter comes into contact with the oven heat. It does expire.
If you're curious about other leaveners, such as eggs and sugar, take a look at this summary by King Arthur.