That's another whole topic though, better left to Cooking Today than ASP Today. Although that doesn't really sound very impressive, you can do some very useful things with cookies. For example, you can make a questionnaire on your website that asks the visitor what their favourite colours and fonts are and then customize the whole look and feel of the website just for them. You can also store passwords so visitors don't have to log in every time they visit.
Cookies do have a few drawbacks.
First, because certain unsavoury elements of the programming community have been known to do nasty things with cookies, most browsers have security settings that can be set so people don't have to accept cookies, so there's no guarantee that you'll be able to use them. If the visitor's computer ever gets a Blue Screen of Death, and has to reformat and reinstall everything, your cookie is gone. Lastly, some legacy browsers can't handle them, though, admittedly, it's pretty rare that you run into a browser that can't handle them. The request is basically treated like a variable. Like any variable, it is important to give your cookies unique names to keep track of them properly.