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The WP Video Lightbox plugin allows you to embed videos on a page using lightbox overlay display.

This plugin can be used to display images, flash, YouTube, Vimeo, iFrame etc in a lightbox overlay. The embedded videos can be viewed on iPhone and iPad too.

Embed a vimeo or YouTube video using shortcode in a WordPress post or page.

Either use words to anchor the link:

click here to open the 'dogs and cats' YouTube video 
or use a thumbnail:
 

How to Turn Off the Related Videos At the End of a YouTube Video

 
If you do not want to show the “suggested videos” that YouTube shows at the end of a video playback then use the following technique:

Add &rel=0 at the end of the video ID which will turn off the related videos display.
Here is another example shortcode that shows you how to use this:

Pirates of the Caribbean Trailer