forum.nginx.org is using https, however the captcha code is currently fetched from http://api.recaptcha.net/challenge?k=6LeSfwUAAAAAAN_HeFPkLBlDvM8PstRDO__CMTTx
Modern browsers refuse to load http content inside https pages, therefore the captcha doesn't show up and registration is not possible (missing captcha code).
It should work if you load the captcha code from https://www.google.com/recaptcha/api/challenge?k=6LeSfwUAAAAAAN_HeFPkLBlDvM8PstRDO__CMTTx instead.
Regards,
Oliver