Postfix is an MTA (mail transfer agent) program that acts as an SMTP server. It is a package in most popular Linux distributions, which makes the installation very simple. Other alternatives include sendmail.
There are plenty of online documentation to help you set up Postfix. For Debian users, http://wiki.debian.org/Manual-Howto (click “Installing and Configuring Postfix on Debian”) is a good resource. It should be mostly applicable to other distributions.
You should also read the documents that come with Postfix. These documents are typically in /usr/share/doc/postfix using a Debian installation. Particularly, start with BASIC_CONFIGURATION_README.gz.
As an SMTP server, postfix receives messages from other SMTP servers. Once it receives a message, it invokes another program to “deliver” the message. In most systems, this means a “mail delivery agent” program (such as procmail) is invoked to store the message into the local file system.