Verify that all database credentials in wp-config.php are correct.
How did you setup your local webserver/database stack? Confirm that it is correctly configured and up and running. Confirm through phpMyAdmin that your DB user exists and has proper permissions for your WP database.
Verify mySQL is listening on port 3306 (the default). If not, the port must be specified in your DB_HOST setting.
What do you have for DB_HOST now? If it’s “localhost”, try “127.0.0.1” instead. Or if it’s already “127.0.0.1” try “localhost” instead. The two should be equivalent, but it’s not always the case.