At Admin Panel -> Settings -> Switches you have a "show subcategories on main page" option. When enabled, each top level category on the front page of your site has a list of subcategories below it. This list of subcategories is cached so that they don't slow the page down much. The cache is automatically refreshed when subcategories are edited, but if for some reason subcategories are altered in a way that doesn't refresh the cache you can refresh it via the regenerate category option on the edit category page, or for all categories by regenerating categories at Admin -> Maintenance -> Regenerate.

To change the apperance, edit the HTML between <!-- BEGIN SUBSUB --> and <!-- END SUBSUB --> in your 'front page' and 'show articles in categories' templates (at Admin Panel -> Themes -> Manage Templates). The category template variables that can be used in this area are restricted to ones which are cached. If you want to use more, use the $subhtmlfields tweak to cache additional fields.