The are a few different reasons why you may have a problem sending emails. 

The following refer to sending problems only, that is, the message never leaves your Outbox.

Rejected messages (bounce backs) are due to different issues, and can be a result of a problem at the sending or receiving end.

The most common cause is that "Authentication" isn't enabled in your email program or device.

All our smtp servers require authentication - this means that your username & password are needed to send as well as receive. outgoing servers. While most email programs will provide a useful message to indicate this (for example in Outlook you would get the message "Error Message - 553 sorry, that domain isn’t in my list of allowed rcpthosts (#5.7.1)", some programs or devices may simply fail. If you have problems sending, you should first ensure that Authentication is enabled. Your program may have an option to enter your username & password again or simply have the option to "use same settings as incoming".

See also:  Configuring SMTP Authentication

Your Internet Service Provider (ISP) is preventing you from sending out on port 25

This is a common problem when traveling abroad. You can check your incoming emails ok, but find you can't reply or send new emails.  See the following article which gives more detail on this issue: "Problem sending email when abroad"

An increasing number of Irish broadband providers are also preventing outbound email from 3rd party servers over their networks. This is generally done by blocking port 25. (you will see which port number your using in the SMTP setting for your email account in your email program or device. Smart Telecom and Magnet are among the better known local ISPs that block port 25, but there are others or sometimes simply parts of networks that are blocked.

To resolve this, you can change the SMTP server setting in your program or device to use the outgoing SMTP server of your ISP instead. When changing, you should generally remove "authentication" from the outgoing SMTP server. A list of Irish ISP's SMTP servers can be found here: Irish ISP SMTP Server list