WordPress keeps your pages, orders and settings in a database. "Error establishing a database connection" means it can't reach that database, so it can't build any page. Your content is usually still there.
How to fix it
For your developer
Test the login from the server, with the values from wp-config.php:
mysql -h DB_HOST -u DB_USER -p DB_NAME -e 'select 1'
To repair tables from WordPress, add define('WP_ALLOW_REPAIR', true); to wp-config.php and open /wp-admin/maint/repair.php. Anyone can open that page while the line is there, so remove it straight after. With WP-CLI, wp db check and wp db repair do the same.
We match the error on the markup of WordPress's own database error page, see WordPress failure signatures.
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