*This article assumes you are already logged into CPanel*
In cpanel you will see an icon labeled as MySQL Databases. Click on it.
This will take you to a new page where you will see:
"New Database: "
Simply enter the name you want to assign to your new database and click on "Create database"
Your new database will now be created and the full correct name of the database will be in the format of "cpanelusername_databasename"
Now, in the Username and Password boxes enter a username and password which will be used to access the MySQL database. Once you have done so, click on the button labeled as "Create User".
You have created a MySQL database and a MySQL user, so now you must add the user to the database in order to access it.
Go back to the "MySQL database" screen and at the bottom you will see "Add user to database". Simply select the user from the dropdown box that you wish to add to a database, then select the database that you want to add them to in the database dropdown menu and click on the "Add" button and thats it, your done.
The user is now added to the database and you can start using it.
Adding a PostgreSQL database is exactly the same procedure and use's exactly the same information, only you will use the "PostgreSQL database" icon in Cpanel.
Alternatively, if your not too sure what your doing, then you can use either of the database creation wizards which will walk you through the entire process.