This means you are supplying a username and password to the MySQL database and are getting rejected.
1. Are you using the correct username and password in your PHP scripts? By default, MAMP uses user 'root', with a password of 'root'.
2. If you are using a CMS, you set up access to the database when you set up your site. Make sure this is the same as the your username and password you are currently using. If you have recently transferred your site to your current machine, it is possible that your setup is using a different password and username to access the MySQL database. In a CMS there is usually a config file.