Authenticate against a Drupal Instance
hook_auth() has been gone since Drupal 5.

But it sounds like you're looking to implement something with user_authenticate().

General outline of how to accomplish this (note that this will send passwords as plaintext - use SSL if that's a problem):


Create an API page using hook_menu() for your app to hit.
In that pages callback function, accept the username and password as POST requests (or however you want to build your app) and run them through user_authenticate().
Return the results of user_authenticate() (either FALSE: fail or numeric: pass) as a json string (or whatever you want) and act accordingly within your external app.