This is a fairly broad problem and could be the result of several factors. Please go through the list below and eliminate each possible problem:
Make sure you have your outgoing SMTP Mail Server set to mail.yourdomain.com.
Make sure you have a connection to the Internet when trying to send outgoing mail.
Make sure your domain name is using right name servers.
Make sure you have authentication enabled for outgoing email.
Make sure that all recipients have valid e-mail addresses.
Make sure you are using the FULL email address as your username.
If all of the requirements above are met, there should be no other reasons why you cannot possibly send outgoing e-mail through your SMTP server. The most common problem is that a client will forget that they must set their email client to authenticate before they can send their outgoing e-mail.
Additionally, new clients usually have this problem while their domain name is in the process of being propagated, in which case they must simply wait for propagation to take place.