As this Ubuntu help document states:
Setting up an email server is a difficult process involving a number
of different programs, each of which needs to be properly configured.
First you'll want to setup a Mail Transfer Agent (MTA), choosing either the default Postfix or exim4. This guide will explain how to install and configure Postfix.
Next you'll want to add Mail Filtering to deal with spam and viruses. If you selected Postfix, then follow this guide.
Then setup a Mail Delivery Agent (MDA) to provide an IMAP and POP3 server that you can connect to with an email client. Dovecot and Courier are two common choices for Ubuntu - see those links for how to install and configure them.
After you install and configure the above, you'll need make sure the following ports are available to the public:
- IMAP: 143
- POP: 110
- SMTP: 25
After that, you'll need to configure your DNS at GoDaddy (if you're not running a DNS server yourself). Here is an article on adding and editing MX records there. You'll also need to make sure you have CNAME records for POP, IMAP, and SMTP. Here is a guide on how to add and edit CNAME records there.
Last but not least, you'll want to make sure you have Reverse DNS setup properly as that can be used as a spam filter. Here is an article on how to add and edit SPF Records there.
As you can see, running a mail server is not a simple task. I'd strongly suggest adding an email package to your domain at GoDaddy, and let them do all this work for you. You can search in Google for promotion codes to make this relatively inexpensive.