Private pages are created by turning on Page Level Permissions. After this is done you can make a particular page private and then you can share it with selected individuals. A private page will appear in the sidebar navigation only to people the page has been shared with.
Page-level permissions allow you to set different levels of access for different people on different pages. For example, you could allow all your friends to see one set of pages, allow your family to edit another set of pages, and then keep yet another set of pages private only to yourself.
To give a user a specific permission on a page you first need to invite that user to the site, once you invite the user to the site you can then grant specific permissions on individual pages for that user. Pages can also provide different visibility levels but you can only make a page less visible than its parent or the site. The site navigation also respects the page-level permissions so if set a page's visibility to Private, on a public site, the page will not appear in the navigation to the public but only to those who can see the page.
More instructions on page-level permissions are on Steegle's site here.
Here Is a way of making a customized sign-in page.
Another way to have private pages is to create a separate site and make it private and share it with the people with whom you want to have access.
Either way you do it, the people must have Google accounts in order to access the private areas.
Another way is to create a Google group, invite the members to it and give the group access to the private pages.
Information on doing that is on Phil Ridout's site here.