Extend page routing with custom parameters

In your case you need to register custom post type for your projects. You can do it by using register_post_type function:

add_action( 'init', 'wpse8170_register_post_type' );
function wpse8170_register_post_type() {
    register_post_type( 'wpse8170-projects', array(
        'label'   => 'Projects',
        'public'  => true,
        'rewrite' => array( 'slug' => 'project' ),
    ) );
}

Don’t forget to flush rewrite rules.

After you register your custom post type, your projects will have /project/%wpse8170-projects% (%wpse8170-project% will be replaced by project slug) permalink structure and you will be able to use single-wpse8170-projects.php template for your projects.