A small string of information sent by a web server to a web browser that will be sent back by the browser each time it accesses that server.
Cookies were invented by Netscape to make it easier to maintain state between HTTP transactions. They can contain any arbitrary information the server chooses to put in them.
The browser limits the size of each cookie and the number each server can store. This prevents a malicious site consuming lots of disk space on the user's computer. The only information that cookies can return to the server is what that server previously sent out.
After using a shared computer, e.g. in an Internet cafe, you should remove all cookies to prevent the browser identifying the next user as you if they happen to visit the same sites.