How do I go about getting the database name?
We can't tell you this information.
Your database name, and the database username / password etc are usually provided to you by your host, or you create them yourself.
Most hosts provide some sort of SQL assistant, like phpMyAdmin, where users can create their own databases, and database 'users'. (each 'user' will have a username / password, you grant him access to the database, and use that username/password on the installation screen).