An operating system is the software that manages the hardware and software resources of the Raspberry Pi and provides common services for the applications.
If asking a question about a specific operating system, such as Debian or Arch Linux, please tag appropriately.
Official Operating Systems
Below is a list of officially supported operating systems and a link to their download and torrent.
NOTE As of July 18, 2012, Raspian Wheezy has replaced Debian Squeeze as the recommended operating system. For more information read this article from the Raspberry Pi Foundation.
NOTE The default passwords for each of the official operating systems can be found password tag wiki.