http://img.youtube.com/vi/<insert-youtube-video-id-here>/0.jpg
http://img.youtube.com/vi/<insert-youtube-video-id-here>/1.jpg
http://img.youtube.com/vi/<insert-youtube-video-id-here>/2.jpg
http://img.youtube.com/vi/<insert-youtube-video-id-here>/3.jpg

http://img.youtube.com/vi/<insert-youtube-video-id-here>/default.jpg

http://img.youtube.com/vi/<insert-youtube-video-id-here>/hqdefault.jpg

http://img.youtube.com/vi/<insert-youtube-video-id-here>/mqdefault.jpg

http://img.youtube.com/vi/<insert-youtube-video-id-here>/maxresdefault.jpg

<script type="text/javascript">
    function youtubeFeedCallback(json){
        document.write('<img src="' + json["data"]["thumbnail"]["sqDefault"] + '">');
    }
</script>
<script type="text/javascript" src="http://gdata.youtube.com/feeds/api/videos/gzDS-Kfd5XQ?v=2&alt=jsonc&callback=youtubeFeedCallback"></script>

$.getJSON("http://gdata.youtube.com/feeds/api/videos/gzDS-Kfd5XQ?v=2&alt=jsonc&callback=?", function(json){
    $("<img/>").attr("src", json["data"]["thumbnail"]["sqDefault"]).appendTo("body");
});

<?php
$json = json_decode(file_get_contents("http://gdata.youtube.com/feeds/api/videos/gzDS-Kfd5XQ?v=2&alt=jsonc"));
echo '<img src="' . $json->data->thumbnail->sqDefault . '">';

http://i3.ytimg.com/vi/SomeVideoIDHere/0.jpg

//grab the default thumb
$attrs = $media->group->thumbnail[1]->attributes();
$thumbnail = $attrs['url'];
    $thumbnail = substr($thumbnail, 0,-5);
    $thumb1 = $thumbnail."default.jpg";
//grab the third
    $thumb2 = $thumbnail."2.jpg";
//grab the fourth.
    $thumb3 = $thumbnail."3.jpg";

//Using simple  cURL to save it your server. You can extends the cURL below if you want it fancy just like the rest of the folks here.
$ch = curl_init ("$thumb1");
curl_setopt($ch, CURLOPT_HEADER, 0);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_BINARYTRANSFER,1);
$rawdata=curl_exec ($ch);
curl_close ($ch);
//using fwrite to save the above
$fp = fopen("SomeLocationInreferenceToYourScript/AnyNameYouWant.jpg",'w');
//write the file
fwrite($fp, $rawdata);
//and then close it.
fclose($fp);