The basics of the Protected Node installation are easy:
Once you have installed the module, you probably want to change the permissions. Permissions allow your users to access the Protected Node functionality based on their roles. Please, see the point about Permissions for more information about those.
Now, you're ready to create a protected node. Either create a new node or edit one of your existing nodes and open the Password protect this page field set. In the field set, select on the This page is protected and enter a password (you have to enter the password twice to make sure you typed it right.) Hit Save. You've got a protected node now.
Either open another browser (not another tab!) or log out your account and go to that new page. The browser will be directed to a form asking you for the password. The content of this page can be tweaked (the text can be changed, cancel link, etc.) See the documentation about the Protected Node Form.
The module can also be used with quite advanced settings. For example, Anonymous users can be allowed to create a page with a password; combined with the node privacy by role module, you can prevent a set of users from access protected nodes and those allowed still have to use the password to see the content.