How can I authenticate against a non-drupal database?
My suggestion would be to take a look at LDAP integration or Lightweight Directory Access Protocol modules to see how they handle authentication.

My recollection is that they hook_form_alter the user_login form to add a new validation function that calls out to the LDAP server if Drupal authentication fails.  You should be able to do this for your own authentication.  We modeled a custom authentication on this method, but I don't recall all of the details.

Just remember that you want to use the external authentication in addition to and not instead of Drupal authenticaion so you don't lock out any admins if/when your other site is down.