How to use the Tag description as the title attribute?

Use get_the_terms() and create a custom array:

function wpse_31396_terms_with_desc( $post_id = NULL, $taxonomy = 'post_tag' )
{
    NULL === $post_id && $post_id = get_the_ID();

    if ( empty ( $post_id ) )
    {
        return '';
    }

    $terms = get_the_terms( $post_id, $taxonomy );

    if ( empty ( $terms ) )
    {
        return '';
    }

    $list = array ();


    foreach ( $terms as $term )
    {
        $list[ $term->term_id ] = array(
            'url' => get_term_link( $term, $taxonomy ),
            'name' => $term->name,
            'description' => $term->description
        );
    }

    return $list;
}

Sample usage for the functions.php:

add_filter( 'the_content', 'wpse_31396_add_terms_with_desc' );

function wpse_31396_add_terms_with_desc( $content )
{
    $tags = wpse_31396_terms_with_desc();

    if ( '' === $tags )
    {
        return $content;
    }

    $taglist="<h2>Cross Reference</h2><ul class="taglist-with-desc">";
    foreach ( $tags as $tag )
    {
        $desc = empty ( $tag['description'] )
            ? '' : '<div class="tagdesc">' . wpautop( $tag['description'] ) . '</div>';
        $taglist .= sprintf(
            '<li><a href="https://wordpress.stackexchange.com/questions/31396/%1$s">%2$s</a>%3$s</li>',
            $tag['url'],
            $tag['name'],
            $desc
        );
    }

    return $content . $taglist . '</ul>';
}