Exim is a mail transfer agent. It receives messages, usually by SMTP, figures out what to do with them according to its configuration and transfers them to another location based on that information.
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Debian has a few packages for Exim. At the time of writing the current major version is 4. The key packages to install are exim4-base, exim4-config and one of the two exim4-daemon-? packages. If you are installing a brand new server, then you can install Exim as a task during the Debian installation stage (see The perfect network server for information on that). If you already have a Debian server then you can install it using tasksel install mail-server or with your favourite package manager.