Do not show related YouTube videos when using the oembed feature of WordPress - put the code into functions.php of your theme or into a custom plugin.

functions.php

<?php
/**
 * Youtube-Video verbieten, related vids am Ende anzuzeigen.
 */
add_filter( 'oembed_dataparse', 'this_add_youtube_norel', 11, 3 );
function this_add_youtube_norel( $return, $data, $url ) {
    if( false !== strpos( $return, 'youtube.com' ) )
        return str_replace( '=oembed', '=oembed&rel=0', $return );
    else
        return $return;
}