Filter a few parameters into WordPress YouTube oEmbed requests. Enable modest branding which hides the YouTube logo. Remove the video title and uploader information. Prevent related videos from being shown once the video has played.

functions.php

<?php
/**
 * Filter a few parameters into YouTube oEmbed requests
 *
 * @link http://goo.gl/yl5D3
 */
function iweb_modest_youtube_player( $html, $url, $args ) {
	return str_replace( '?feature=oembed', '?feature=oembed&modestbranding=1&showinfo=0&rel=0', $html );
}
add_filter( 'oembed_result', 'iweb_modest_youtube_player', 10, 3 );