How do I restore a site from backup
If you've copied the Craft files to a new place on the server, then you can either leave the database connection information the same and have it keep on pointing to the current database instance, or edit craft/config/db.php (or your .env file if that's how you're setup) to point to a new database instance.

If the new one is a blank database, the Craft installer will want to run.  If you want a copy of your old database on the new one, you'll need to do a database dump from the old one and restore it to the new one.  If you do the latter and you're still getting Craft to prompt to run the installer, then there is something wrong with the database connection settings between Craft/PHP and the database and those should be adjusted.