As I understand it, there are 2 options. The first is to include moodle's config.php file in your web application - This will give you access to the Moodle session, so you can determine if a user has logged in to Moodle, although it also defines a lot of functions and globals, as well as connecting you to Moodle's databsae, so there are security issues to consider.
Another option is to use MNet to make your web application appear to Moodle as another Moodle instance. This way you can have users logged in to Moodle roam to your web app, and vice versa. This is similar to how Mahara allows single sign-on with Moodle.
Please note that I've never implemented either approach above so I don't speak with authority, but they might be a good place to start.