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Relaying means that you are trying to send a message through a mail server without being logged in to that server, and the people who run the server have chosen to disallow that to prevent spammers from using the server.
Example. You have an account on Comcast and you also have an account via your domain hosted on pair.com. You cannot use the SMTP for your pair account to send via your ISP host Comcast. This is relaying. You can use your Comcast SMTP to send your pair mail.
You want to prevent unauthorized senders -- in other words, spammers -- from using your server as an SMTP relay to hide the real origin of their messages.
Links for more info on relaying: