How do I add Widgets to my theme and activate them on my site?

It sounds like you’re trying to add a new widget area (dynamic sidebar) to your Theme.

That process has three parts:

1. Register the dynamic sidebar in functions.php, using register_sidebar():

   function wpse121723_register_sidebars() {
       register_sidebar( array(
           'name' => 'Home right sidebar',
           'id' => 'home_right_1',
           'before_widget' => '<div>',
           'after_widget' => '</div>',
           'before_title' => '<h2 class="rounded">',
           'after_title' => '</h2>',
       ) );
   }
   add_action( 'widgets_init', 'wpse121723_register_sidebars' );
   

   This registers the dynamic sidebar with WordPress, which displays its UI in Appearance -> Widgets in the Admin. The important part here is the id parameter, which you’ll use in the next step:

2. Output the widget area (dynamic sidebar) in your Theme template, wherever appropriate, using dynamic_sidebar( $id ):

   <?php dymamic_sidebar( 'home_right_1' ); ?>
   

   This actually displays the dynamic sidebar in the template.

3. Populate the dynamic sidebar, via Appearance -> Widgets in the Admin.

It sounds like you’ve done #1 and #3, but not #2.