You basically have two choices:
cookies
passing a session id through the url
Either way, it'd be implemented pretty much teh same. It should be noted that PHP's session handling functions are propagated through either cookies or a session id through teh URL too, they just have the added benefit of convienent server-side storage for teh duration of the session.
And no, you can't just set a boolean 'logged in' field to true in MySQL. Though that'll probably be part of your solution.
The problem is that on each request, you need a way to identify which session the request belongs to... thus the need for a unique session id assigned to the browser.