How to detect when a youtube video finishes playing?

This can be done through the youtube player API:

http://jsfiddle.net/7Gznb/

Working example:

    <div id="player"></div>

    <script src="http://www.youtube.com/player_api"></script>

    <script>

        // create youtube player
        var player;
        function onYouTubePlayerAPIReady() {
            player = new YT.Player('player', {
              width: '640',
              height: '390',
              videoId: '0Bmhjf0rKe8',
              events: {
                onReady: onPlayerReady,
                onStateChange: onPlayerStateChange
              }
            });
        }

        // autoplay video
        function onPlayerReady(event) {
            event.target.playVideo();
        }

        // when video ends
        function onPlayerStateChange(event) {        
            if(event.data === 0) {          
                alert('done');
            }
        }

    </script>