Hi, I tested it by myself before giving you the instruction. I didn't write it from memory.
Suppose, The URL of the Youtube video you want to display is - http://www.youtube.com/watch?v=66TuSJo4dZM
If you go to that page you will see some buttons right underneath the player. For example-
"Like", "Add to", "Share". Click the "Share" button (please see the screenshot)-
http://my.jetscreenshot.com/8105/20110923-vtrb-52kb.jpg
Once you click the "Share" button the "embed" button will show up. Click "embed" (please see the screenshot) -
http://my.jetscreenshot.com/8105/20110923-b6sl-41kb.jpg
Now you will be able to see the embed code (between <iframe>....</iframe> tags) with additional checkboxes (please see the screenshot) -
http://my.jetscreenshot.com/8105/20110923-bzei-64kb.jpg
By default "Show suggested videos when the video finishes" option is checked. If you uncheck it you will get a URL with rel=0. Now you can use this URL with the colorbox shortcode I mentioned before.