Different web hosts have different methods of creating databases, which are documented here. …
Most will prepend your hosting account username as a prefix to your chosen database and user names. Upon successfully creating a database, most will show you the full database name (with prefix) and one or more user names (with prefix) and passwords. Write this information down (including the prefixes) for future reference.
Best practice is to use one database per application and version and that each database name indicate the application name and release (e.g. joomla15). If circumstances force you to use a single database for multiple versions of an application and/or multiple applications, then use a nondefault table prefix to group tables by application and version (e.g. jos10_ vs. jos15_).
Open cPanel. Click on "MySQL Database Wizard" and follow the instructions to set up a new MySQL database, user, and password. Upon creation you are asked what privileges should be granted the new user. TODO: Document which privileges are really needed for common scenarios.
In the Account Control Center (ACC), go to Database Management – Create a New Database and follow the instructions. Pair will create three new MySQL users for each database: a full access user, a read/write user, and a read-only user, and assign each a different password. You may change the passwords manually. Print out the confirmation screen for your records.