Session variables are essentially stored in temporary files server side as far as I'm aware and really cookies don't come into it.
Judging by your previous posts, this is PHP. Assuming that:
You're starting each page with session_start();
You're using $_SESSION to access files.
You may need to ensure you're closing every single tab/instance of the browser you have open in order for the session to be closed.
You might also want to check out your php.ini configuration file as it's possible that your session IS being stored in cookies, in which case you need to set the value of your "cookie-lifetime" to 0.
PHP: Runtime Configuration - Manual