The MUA is the program that a user uses to read his or her mail. Some examples: pine, mutt or Netscape (on Windows Outlook). The MUA can use several protocols to fetch the mail. The most used services are POP3, IMAP, but SMTP is also possible.
All incoming mail for a user is per default stored in one file in /var/spool/mail/<username>. You could read your mail with more or less, but that is not very comfortable.
Another option is to use a mail directory called Maildir in a users home directory to store the mail. This is often used for IMAP.
The most secure way for fetching your mail is by using pop3 or IMAP over SSL.
apt-get install courier-pop-ssl
The configuration can be found in /etc/courier/.
[META] needs more docu about interfaces to run on and protocols/ports to use.