In this post to the CodeOfANinjs.com site, they walk you through password hashing, salting and storage using the PHPAss tool from OpenWall. The post itself is a bit older, but the content still provides a good example to teach the basics.
I think the main reason why we have to hash passwords is to prevent passwords from being stolen or compromised. You see, even if someone steal your database, they will never read your actual or cleartext password. I know that some PHP frameworks or CMS already provide this functionality, but I believe that it is important for us to know how its implementation can be made.
The tutorial shows you how to use the library and how to store the result in a simple "users" table in a MySQL database. The examples hash the password given from a simple form and use prepared statements (via PDO) to save it to the database. All PHP, HTML and CSS code you'll need - including the login form that checks the username/password - is included. There's also a few screenshots showing what the resulting forms and data should look like.