Of course you can. Just set the video to Unlisted and now only people with the link to video can see.
Lots of reasons why you would want to do that. I make videos to show only my list, for example. You may want to hide old content but not delete it entirely. How many fingers do you have? You ought to be able to come up with at least that many reasons.... lol
I have found private YouTube videos to be an excellent way of sharing ideas with other members of my mastermind group.
Yes, you can make a video "private" on YouTube. When you upload a video, there is a box on the top right of the upload page where you set the Privacy and Publish Settings. You can either chose to mark the video "Private" and only a select people in your YouTube followers/friends can see the video. You can mark the video as "Unlisted", this means the video will not show in the YouTube search and only people given the direct link to the video will see it. Hope that helps. Have a good day!