I would like to have a server that is able to receive emails. Then I want to use PHP to program the way the emails are shown to the users. Can I do it purely with PHP? I mean, it is not a problem to send emails from PHP but I do not know if I can receive emails by PHP? (In a way PHP receives POST requests).
ADDED
As a response to the first answer, I would like to specify that it looks like I need an SMTP server. I want to be able to communicate with the SMTP server in a programmatic way. For example, I want to have a possibility to "tell" to the SMTP server to create a new e-mail address. I also need to know where incoming emails stored and in what format. For example, how I can extract the "sender", "cc", "bcc" from the file corresponding to the received mail.
If you are writing it from scratch then you'll need the specification for SMTP. I would advise very strongly against this. SMTP servers are hard to write, and there are several really good open source solutions out there.
My understanding of PHP is that it does very poorly when it comes to multithreading, so it probably isn't a good solution for this problem.
Servers that receive mails do not typically show them to users. They usually store them in a standard way (such as Maildir or mbox) which other software (such as a local email client or an IMAP server) accesses.
The job of showing email to a user is belongs to email clients. Web based PHP web mail software includes SquirrelMail and RoundCube. AFAIK they both act as IMAP clients. See the IMAP specification.
Pick an SMTP server that runs on your OS. Read the instructions to find out how to configure delivery and accepted addresses. It usually comes down to manipulating text files.
Again. See the manual for the mailserver. Most will give you options about where to store the data and in what format.
Then you just need to decide if you are going to get PHP to dig into those directly, or use an IMAP server in between.