would like to have a sever that is able to receive e-mails.
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.
Than I want to use PHP to program the way the mails are shown to the users
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.
As a response to the first answer, I would like to specify that it looks like I need a SMTP servers. 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.
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.
I also need to know where incoming mails stored and in what format. For example, how I can extract the "sender", "cc", "bcc" from the file corresponding to the received mail.
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.