Your Elgg installation is located in your root directory from your public_html. This means it is installed in /, not /elgg. Either that or htaccess is rewriting / to /elgg.
When I navigate to your website, I see that the Elgg software cannot make communication with the database. You should have created a MySQL Database, User, and assigned to User to the Database, and then inputted the information when you installed Elgg. Since I think Elgg has already performed the installation progress, code edit /public_html/engine/settings.php with the correct database information:
Hostname: localhost
Username: CPANELUSER_MYSQLUSER (your cPanel username, then an underscore, then your MySQL username)
Database: CPANELUSER_MYSQLDB (your cPanel username, then an underscore, then your MySQL database)
Password: MYSQLPASS (your MySQL password)