You need to have a facebook app, then in your site use the facebook js sdk.
You always have different sessions per site, you'll need to make a session for yours, you don't use the facebook session, just the sdk.
When the user loads your page you can use the FB.getLoginStatus to check if the user is logged into facebook and is authenticated with your app, if so then you get the needed user data and can then log the user into your site.
If the user is not logged into facebook, or hasn't authorized your app then you can use the FB.login method to log the user in, when the user comes back from that process you should have the data you need in order to register/log him into your site.