Kiki asks: ""but if I move to another website (even I do not close the browser), the session ID will be expired?"
Not automatically, at least not that quickly. Sessions do expire according to a value set in php.ini, but the default is 4 hours. In my BBS session code, I ask users to log out (which is reasonable, since I've asked them to log in!) and when they do, I call session_destroy() to wipe out the session records.
As Scott points out, sessions can be implemented using either cookies or URLs to hold the session IDs. (The actual session variables -- the part you as the programmer are interested in -- are actually stored on the server, by default in special temporary files. The session IDs are used to locate the correct set of session variables.)
Let's imagine that a site doesn't use cookies and instead propagates the session ID using URLs.
If you leave such a site without logging out, the session won't be destroyed. If you return to one of the URLs that contains a session ID, your session will pick up where it left off.
But if you return to the "front door" of the site, or a bookmark with no session ID or an old session ID, you will have no way to reclaim your session variables. So, while the Web server will still have your records, it won't recognize you.