I am thinking that the message is something like "Relay access denied". Am I right?
If it is so, again, there can be a few reasons.
Best is, search the forums for "Relay Access Denied", because this topic has been discussed a number of times and you will be pleased by the depth in which people have already discussed it.
If none of these help you, please do post back.
(For starting, the tip is: create another user in Zimbra and send email from one user to another user
both of which are in the same instance of Zimbra. Thing to notice is whether the email is transmitted or it also returns back with error).