There was an official way to do this, from the YouTube folks.
(link to YT official way - NB: now broken, but YT hasn't (bothered to) correct this page: http://apiblog.youtube.com/2009/02/youtube-apis-iphone-cool-mobile-apps.html)
In July 2010, they deliberately removed that - it no longer works (I've got an app that worked fine prior to the change, and now doesn't, using YT's own code).
There's a "new" way that the YouTube folks described as "experimental" - but it's only partially working. Where you could previously put YT videos into your iPhone GUI, now you cannot - they are "required" to be given an entire screen to themselves.
(link to YT official "new" way: http://apiblog.youtube.com/2010/07/new-way-to-embed-youtube-videos.html)
(link to YT official thread, that ends with "it might work... eventually": http://code.google.com/apis/youtube/forum/discussion.html?place=topic%2Fyoutube-api-gdata%2FtMYvulpGUow%2Fdiscussion)
Worse, YouTube engineers didn't provide any user feedback; when the user clicks on a button, you have to wait for the YT servers to process a couple of web lookups (you can see this if you look at your console log files).
All the user sees is: "this app appears to have crashed".
Argh! Is there any other way? What are other people using here that's working still - on OS 3.x + 4.x - since the recent YT changes?