Follow-up Comments Notification Without a Plug-in

It’s quite simple, just hook a function on comment_post in which you check if the comment is a reply and send the author of the parent comment an email:

add_action('comment_post', 'notify_author_of_reply', 10, 2);

function notify_author_of_reply($comment_id, $approved){
  if($approved){
    $comment = get_comment($comment_id);
    if($comment->comment_parent){
      $parent_comment = get_comment($comment->comment_parent);
      wp_mail($parent_comment->comment_author_email, 'Hello', 'Dude you got a reply...');
    }
  }
}

For handling unsubscriptions, you can add store emails of people who don’t want notifications inside an option (and check if $parent_comment->comment_author_email is not within that list in the function above).