How can I link my private videos on YouTube to be played on my website. The point is to bring more viewers to my site by making videos available on my site, yet private on YouTube.
You can embed private videos, but private videos can only be shared with up to 50 users. Other people won't be able to view it.
Making your video "unlisted" will mean other YouTube users won't find it in YouTube search results, via your user page or similar videos. This is what you want to be using if you want to share it on your site but keep it out of YouTube's public eye.
But, I people can still share that link once they've seen it. It's not entirely locked down.
These are the only choices at the moment, though.
There's a site called Viddler which gives an alternate solution. You can take a look at it. They have various paid plans but also an ad-supported free plan. This article goes into the details of how you can setup the solution.
You can also read its mention in our directory.
You can choose the "unlisted" option, so that your video doesn't appear in search results, but anyone with the link can view it. This allows you to embed the video. However, the URL and embed code for the video will show up within the player when the video stops playing.