In order to move phpBB2 to your public_html directory, first you have to move your files from the subfolder to the public_html. The easiest way to do this is by using an FTP client manager like FileZilla.

The next step is to modify the phpBB2 configuration from your cPanel > phpMyAdmin. Select the phpBB2 database from the drop-down menu on the left. Then open the SQL tab and paste the following query in the text field:

UPDATE phpbb_config SET script_path = 'new_relative_path_to_phpbb' WHERE config_name = 'current_relative_path_to_phpbb';

You have to change the 'new_relative_path_to_phpbb' and 'current_relative_path_to_phpbb' in the query shown above to meet your environment. For example, if you move the phpBB2 installation from subfolder "forum" to the root of your web hosting account (the public_html folder), the query will look like:

UPDATE phpbb_config SET script_path = '/' WHERE config_name = '/forum/';