Firstly try sending an email to yourself (so send a mail from eM client to the address you have setup in eM client) to see if you receive that back, if you do then your mail settings in the client are correct.
if you are successful in sending a mail to yourself then check with whoever is hosting your domains emails that they have setup a few things to ensure that your emails will be delivered to everyone, firstly you will require reverse DNS (otherwise known as a PTR record) on your mail server, secondly it's a wise move to ensure that your mail servers DNS entries have a SPF record (Sender Policy Framework record) associated with them.
without both the above in place a lot of the larger mail organisations, such as hotmail, Gmail, AOL, etc will not accept emails from your domain.
It all sounds very complicated I know, but as long as the first test of sending a mail to yourself works then the next steps are something you will need to take up with the support people for whoever is hosting your email.