Hello Rubén
The one and only reason that this particular message is displayed on the site is when the mail server (The one that you configured in Joomla's Global Configuration) cannot contact the recipient e-mail address.
Normally to debug such issues, I would perform the following checks first:
1. Check if your using a proper email settings in Joomla Global Configuration. If unsure, check with your hosting provider.
2. Is the recipient's e-mail address valid?
Regarding the cronjobs question, hope this can help you.
Why do we need cronjobs?
Because processing emails are always slow. When there are hundreds of emails to be processed at a time, the server needs to contact the respective mail servers to deliver the mail. However, the process for each delivery might differ from time to time and this would result in higher bounce rate of users on your site.
Consider this scenario:
John posted a comment on an update which has 10,000 previous comments. To deliver 10,000 emails as soon as you post a comment would take more than an hour. Would you want your user's to wait?