First, Drupal adds "menu-" to your custom menu. So if you named your menu "custom-menu", when referencing your custom menu you have to call it "menu-custom-menu".
The drupal api call to make is "menu_navigation_links", and then pass the result to "theme" to render the list.
print theme('links', menu_navigation_links('menu-custom-menu',0));
Look how the code above is similar to what you're used to doing for the $primary_links, and $secondary_links that are built into drupal, as shown below:
print theme('links', $primary_links, array('class' => 'links primary-links'));