There are a few methods suggested for doing this online, but none of them seem to work.
For example:
http://blog.makezine.com/archive/2008/11/youtube-in-720p
You can do this by adding a parameter &hd=1
to the video URL. That forces the video to start in the highest resolution available for the video. However you cannot specifically set it to 720p, because not every video has that hd ish.
http://www.youtube.com/watch?v=VIDEO_ID&hd=1
http://code.google.com/apis/youtube/player_parameters.html
UPDATE: as of 2014, hd is deprecated https://developers.google.com/youtube/player_parameters?csw=1#Deprecated_Parameters
This is an embed example of video played in HD 1080.
<iframe width="560" height="315" src="http://youtube.com/v/IplDUxTQxsE&vq=hd1080" frameborder="0" allowfullscreen="1"></iframe>
Let's break apart the code:http://youtube.com/v/
video_id
&vq=hd1080
Video id for that video: IplDUxTQxsE you will see this type of random code in the link of every YouTube video.
So far so good, this trick works for playing full HD videos directly on webpages!
You can change the quality to 720 too. &vq=hd720
In case you're still wondering how to do it, then add: &feature=youtu.be&hd=1 Actually now I checked, this works only when you're sending the URL to someone else, not on embed.