My work specializes in video/audio, mostly for tv/radio but more and more for the web. I have thousands of hours of experience with flash video, mostly incredibly frustrating until I started using swfobject exclusively. The way I use it is to create an html element like a DIV and replace the div with the flash swf, and style the div with css.
Just find the video you want on YouTube, press the embed button with privacy mode, and grab the url.
<object width="480" height="385"><param name="movie" value="http://www.youtube-nocookie.com/v/-HIB706Tvwo?fs=1&hl=en_US"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube-nocookie.com/v/-HIB706Tvwo?fs=1&hl=en_US" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object>
This is essentially the same for all methods other than the BETA iframe.
Just last month Youtube/google announced a new way to embed flash, and it's the absolute easiest yet. It uses iframes, so you should take the time to add the iframe using DOM scripting rather than just add it like this (which works):
<iframe title="YouTube video player" class="youtube-player" type="text/html" width="380" height="190" src="http://www.youtube.com/embed/-HIB706Tvwo" frameborder="0"></iframe>Advanced Youtube Embedding
Flash Video Player
The method I use now on this site involves one additional piece of software, a flash video player for the web. Their are many different flash video players out there, youtube has it's own video player, but the one that I use is the same one I have been using since.. oh.. 2006 because it's amazing. No matter which video player you choose, you want it to basically just be an .swf file that loads the videos, whether its
.f4v, or streaming. Currently (10-2010) JW FLV Player is free (sorta) and the source code is open (for now) and is called JW-FLV-Player. Grab it before it goes premium only. I like to use and keep older versions, like I have a very old version that I used to use to play mp3's from a playlist. Actually it's still online if you want to listen..
Using this software you are able to play youtube videos off the youtube servers with a flash player on your server. Here's an example of mine that uses a simple rss playlist of youtube video urls to play the videos, you, the reader, are encouraged to view the source code to figure it out. ;)