I would like to incorporate phpbb logins with the rest of my site.
I know about using the logins and passwords in another script but I would like to make it so that I can give the same permissions

MOD, Admin, and the average user but I am unsure of how this is done.

Use phpbb's member tables for all your membership and just use the session_begin() function from phpbb to login users for your site. Permissions and what-not are much more complicated to transfer over to your site. Dont forget to configure phpbb's cookie settings to .yourdomain.com and path = '/'.