For privacy reasons, users who upload videos on YouTube sometimes opt to set their videos as unlisted or private. While you can watch an unlisted video if someone shared its link with you, accessing a private video requires you to contact its owner and request permission to watch it.
While anyone can watch unlisted videos, they do not appear in YouTube's search results or in the channels of the users who uploaded them. Because you cannot rely on YouTube's search engine to find unlisted videos, you can only watch them if someone shared their direct link with you. To watch an unlisted video, you should therefore consider either sending a message to the uploader or contacting one of your friends who has already viewed the video and asking that person for the link.
By default, only the user who uploaded it can watch a private video. However, uploaders can also manually add users to a list of people allowed to access a private video. If you want to watch a private YouTube video, you should therefore contact its uploader, provide her with your Google account username -- which YouTube needs to authenticate you -- and ask her to grant you access to that video.