This is from the Google's Productforums
For clarification. Yes, you can embed private videos. However, it really isn't feasible to do so since YouTube's private videos require that the viewer have a YouTube account, be signed into that account and have that account added to the video's access list. They are limited to 50 people.
You can embed unlisted videos, they don't have the same authentication restrictions as private videos. However, all that is necessary to view an unlisted video to know the URL. If your website is crawled by search engines, the unlisted video may as well be public.
You may want to look into self-hosting your own videos to have better control and to be sure that you aren't violating YouTube's Terms of Service (which prohibits some uses of the embedded player).
Vimeo.com also has more robust access controls and can only allow the video to be embedded on a specific domain. However, they don't allow commercial use...so that may be a catch if that applies to you.