Adding to the new "FAQ":
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It looks like a problem with your MySQL username, password, database, hostname, or any combination of the two.
Are you sure you have created a database, set up a username and password, and also (on some servers) added the user to have permission to the database?
Most hosts use "localhost" as the hostname, but your host could use something else. This could be mentioned in your hosting FAQ or your MySQL control panel / setup page.