BuddyPress.org

This is the original registration form from bp-default theme https://buddypress.trac.wordpress.org/browser/tags/1.5.5/bp-themes/bp-default/registration/register.php

Back up your current registration/register.php then download register.php from link above and upload to your theme’s /registration/ folder. If that doesn’t work, I suggest deactivating BuddyPress and BP plugins then changing to Twenty Eleven theme. Check if registration is working on WordPress alone.