You can receive “553 SMTP Relaying Denied” error messages when trying to send an email for a couple of reasons.
The first reason is that you are trying to send an email to an address not recognised by the server. In other words, the email address is misspelled or doesn't exist.
The second possibility relates to your authentication. If you are sending an email outside your own domain (ie emailing someone who’s email address does not end with the same @domain.com suffix as your own), you need to authenticate first using your username and password.
When troubleshooting emails it's always a good idea to analyse the bounceback email header information, and research based on what you find.