Asterisk is an open source/free software implementation of a telephone private branch exchange (PBX) originally created in 1999 by Mark Spencer of Digium. Like any PBX, it allows a number of attached telephones to make calls to one another, and to connect to other telephone services including the public switched telephone network (PSTN). [wikipedia.org]

Trixbox and Elastix are the two famous distributions containing the Linux operating system and several other utilities that facilitate the user in configuring the phone exchange through a graphical user interface.

Elastix is an open source Unified Communications Server software that brings together IP PBX (Asterisk), Email, IM, Faxing and Collaboration functionality. It is known for having an easy-to-use Web interface and including capabilities such as a Call Center software with predictive dialing. The Elastix functionality is based on open source projects as Asterisk, HylaFAX, Openfire and Postfix. Those packages offer the PBX, Fax, Instant Messaging and Email functions, respectively.

Asterisk Appliance is a hardware based on either Asterisk and AsteriskGUI, or the distributions like Trixbox or Elastix.