This could be the infamous 'Port 25 relay' problem. Some service providers will not allow their email servers to be used for outgoing mail if that request did not originate from their own network. This is because, for Port 25 mail, there is no authentication required and so the service provider cannot be certain that you are who you say you are. Although incoming mail - POP - requires a password, the 'fathers' of the Internet, when they put together the protocols for outgoing mail, did not consider that a password would be required for outgoing mail. After all, you need access to your own secure surface mail letterbox to be able to retrieve your paper letters, but anyone can drop an envelope into a mailbox.
But there have been many instances of people sending emails claiming to be a person they were not and so most service providers will not allow sending of emails from their server if the person sending is not logged into their home network. So the first test is to see whether you can send from your home network. If you can, then it's the Port 25 relay problem and there are ways around this, so get back to us.
If you can't send from your home network, then it's likely one of the problems discussed by the other posters.
Tim