Here is how you configure automatic (security) updates on Debian:

aptitude install unattended-upgrades

Here you can configure it:

nano /etc/apt/apt.conf.d/50unattended-upgrades

This could look like this:

Unattended-Upgrade::Origins-Pattern {
        "origin=Debian,archive=stable,label=Debian-Security";
};
Unattended-Upgrade::Package-Blacklist {
// add packages here that need manual steps like this:
//        "vim";
}
Unattended-Upgrade::AutoFixInterruptedDpkg "true";
Unattended-Upgrade::MinimalSteps "true";
Unattended-Upgrade::InstallOnShutdown "false";
Unattended-Upgrade::Mail "root";
Unattended-Upgrade::MailOnlyOnError "true";
Unattended-Upgrade::Remove-Unused-Dependencies "true";
Unattended-Upgrade::Automatic-Reboot "false";
Acquire::http::Dl-Limit "200";

Now create the following file:

nano /etc/apt/apt.conf.d/02periodic

With this content:

// Enable the update/upgrade script (0=disable)
APT::Periodic::Enable "1";

// Do "apt-get update" automatically every n-days (0=disable)
APT::Periodic::Update-Package-Lists "1";

// Do "apt-get upgrade --download-only" every n-days (0=disable)
APT::Periodic::Download-Upgradeable-Packages "1";

// Run the "unattended-upgrade" security upgrade script
// every n-days (0=disabled)
// Requires the package "unattended-upgrades" and will write
// a log in /var/log/unattended-upgrades
APT::Periodic::Unattended-Upgrade "1";

// Do "apt-get autoclean" every n-days (0=disable)
APT::Periodic::AutocleanInterval "7";

Of course the server needs to be able to send mails so it can send mails in case of problems.

You can test it like this:

# mail -s test mail@example.com
My testmail
.
EOT

Hope this helps somebody.

Update: Some updates caused dpkg questions about changed config files and therefore failed.

For example php5-fpm did ask this:

Setting up php5-fpm (5.4.4-14+deb7u10) ...

Configuration file `/etc/php5/fpm/pool.d/www.conf'
 ==> Deleted (by you or by a script) since installation.
 ==> Package distributor has shipped an updated version.
   What would you like to do about it ?  Your options are:
    Y or I  : install the package maintainer's version
    N or O  : keep your currently-installed version
      D     : show the differences between the versions
      Z     : start a shell to examine the situation
 The default action is to keep your current version.
*** www.conf (Y/I/N/O/D/Z) [default=N] ? dpkg: error processing php5-fpm (--configure):
 EOF on stdin at conffile prompt

This caused PHP5-FPM to stop and all PHP sites to show an Internal Server Errror…

So you don’t run into this problem, create /etc/apt/apt.conf.d/local with this content:

Dpkg::Options {
   "--force-confdef";
   "--force-confold";
}

This tells DPKG to keep the old config file. It will create .dpkg-dist files with the package distributer’s version. More information on this can be found here.