A cookie is a small data file that a remote website will place on your computer in order to remember you and your settings (session state) when revisiting at a future point in time.
A common example of this would be a cookie placed by an email web page so that when you close the page and later return your login session will automatically be restored without having to re-enter your username and password, or if you were on a shopping site and placed some items in your shopping cart and then left, upon returning to the page your shopping cart would be restored.