Question: I want to change the root password so it is different than the admin password.

I need to change the password so the admin can still be logged on, but the root password to modify the hosts file through terminal with nano at /private/etc/hosts is different. 
Even if the admin is logged in, he will need a different set of codes to modify the hosts file.

Is this possible?

Answer: 
I'm not sure I get what you want, but here's how things work on OS X.

* You can't prevent an admin user from becoming root with their own password.
* Root user isn't enabled, and doesn't even have a password unless you set one.

So, if you want to prevent a user from editing hosts, you need to remove their admin rights. Anything you can lock with special permissions, that admin user can unlock so what I'm guessing you ask is not possible.
You can change the root password by using the Directory Utility. This application is found in /System/Library/Core Services/

If root is not yet enabled:

1. Click the lock and authenticate with your admin account to make changes
2. Click the Edit menu and Select "Enable Root User"
3. Set a unique password for root

If root is enabled:

1. Click the lock and authenticate with your admin account to make changes
2. Click the Edit menu and Select "Change Root Password..."
3. Set a unique password for root

This will not stop an admin from running sudo however.