BuddyPress.org

It’s one of or a combination of two issues: the folder permissions may not be set correctly (these can normally be corrected via a decent FTP client folders generally need 755 and files 644)

Check the WordPress codex for the correct permissions required

The other issue and more likely the problem here is that the server group hasn’t got the correct permissions to create and write to directories, if you are on shared hosting you may need to contact your host to get this sorted out and also ensure that PHP ‘Safe Mode’ is turned off.