The next time you visit that particular site the server searches your browser's cache for cookies that contain information you may have previously provided to that site. The browser then returns a copy of the cookie to the server. Keep in mind that a browser already gives up information about you without cookies. This information includes your operating system, your browser type, and even your IP address. See below to find out if your browser is giving your out.
Most people are unaware of cookies because browsers' are configured to accept cookies by default.
It is impossible for one site to read a cookie from another site. They can only be used to store information that you have provided at some point. If you are uncomfortable with cookies, you have several options. Netscape Navigator 4.0 users can also disable cookies. You will get many servers that will offer the same cookie repeatedly even after you refuse to accept the first one, quickly becoming quite a nuisance.