Back in the early days of the Internet, most hosts were quite hospitable.
Their administrators were quite willing to "lend a helping hand," allowing
people to use their mail servers freely. Now, alas, things have changed.
Very few systems allow their mail servers to be used by anybody else.
Let me give an example of relaying. Let's say you're using Netcom as your ISP and you want to send a message to somebody on AOL. Instead of using Netcom's mailserver, you decide to use the one used by MIT. This is "mail relaying." If MIT's mailserver is configured to allow this, no problem. If not, you'll get an error message, saying that relaying has been denied.
Why is this done? Good question. By luck, I just happen to have an answer. The most common reason, today, for relaying your email is to make it hard to find out where you sent it from. Why would you do this? Well, if you happened to be sending Unsolicited Commercial Email, commonly called "Spam," you might not want it traced back to you. If it were, you'd probably lose your account. There is another reason that people try this, and I'll deal with it later.
How do the mailservers check this? It's really very simple: the know what IP address you have. They can do what's called a "reverse DNS lookup" on this and find out what service you're logged into. If it isn't one that's in their acceptable list, or you're not sending mail to one of their subscribers, they refuse the connection. Today, most SMTP (Simple Mail Transport Protocol) servers do this. If you're getting this error message, you're probably trying to use a mailserver that's not part of the same ISP that you're using to connect to the Internet.
I mentioned, up above, that there's a second reason to try this. Let's say that you have a home account with Earthlink. At work, you're on a LAN (Local Area Network) with its own , direct connection to the Internet. Let's also say that your corporate LAN doesn't use Earthlink to connect. You want to check your home email through work, and send responses. Alas, if you're using your direct connection to the Internet, your IP address isn't valid for Earthlink, and you can't send email, except to Earthlink members.
How do you fix this? Simple: use the outgoing, or SMTP server specified by the service providing your Internet connection. Thus, if you're company connects through (let's say) pacbell.net, you'd use whatever sever they supply. How you set this depends on your email program. If your email program only allows you to list one server, using both for incoming and outgoing, you'll need to upgrade to a more modern program that allows you to configure different servers for incomong and outgoing servers.