They contain information about when you were at the site, and what parts of the site you viewed. Cookies are used in many ways, and one of the most widely used application of cookies are 'shopping carts'. If you are viewing an online catalog, and you start putting items in your shopping cart, a cookie is what keeps track of what you have selected. If you leave the site without finishing, you can come back later and pick up where you left off, enabling you not retrace your steps. If you fill out a questionnaire, a cookie is used to keep your answers in one place.
Cookies generally expire after a set period of time once it is received.
Another example to explain cookies is a baggage claim ticket. When you arrive at your destination and want your luggage, you have to have your baggage claim. A cookie is like it in that if you return to a site, your computer sends that 'baggage claim' to the web server, letting that server know you were there and who you are. Many negative rumors about cookies are around, including the CIA using them to keep tabs on you, or a way to put a virus on your computer. These simply are not true.