If you run the command:
> bash install_webwork.sh
as root, then it shouldn't ask you for a password up front.
If you run
> sudo bash install_webwork.sh
then it will prompt for a password, which I believe must be the password of the current user. This will only work if that user has sufficient sudoer privileges.
During the installation, it runs mysql_secure_installation which will ask you to set a mysql root password if there isn't one already set. You will need to type that password later in the process to create the webwork database. Later it will ask you to choose a password for the database user, which is a new password that you are creating.
I think that these are all of the passwords involved in the process. Let me know if this doesn't answer your question.
Danny