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.