To upload a private [actually unlisted] video to youtube:

Note that an unlisted video can't be searched for or stumbled on by anyone but *anyone*
who has the link is able to view the video with restriction so consider if you just want 
to obscure it for casual privacy sake or if you really need a secure solution 
then youtube doesn't provide this.

• Open up youtube and sign in (1 below)

• Click upload (see 2 below)

• Click on the "select files to upload" button (see screenshot below)

• "Browse to" and select your file and "open" to upload (see 3 and 4 below)

• While the file is uploading you can amend the title and other required information and if you want this to be unlisted make sure you chose unlisted. (5 below) (Private means that only you can view it no one else so "private" is no good for sharing).
• Once complete people will be ablet to view the video if they have the URL (link) shown (see 6 below)