Apart from many PBX equipments manufactured by many PBX companies, there are few freely available software based PBX systems that run on a PC. Most probably, there will not be a single person who has not heard about Asterisk, the most famous open source PBX system as well as many CTI applications.

Open source PBX systems are supported by large online communities. Many people use these open source PBX systems as they offer excellent customization as well as cheap CTI board integration.

1. Asterisk
Asterisk is supported by Digium Inc. the VoIP company with many CTI boards and applications. Asterisk is software based PBX system that runs on PC with operating system BSD Linux, Linux, MAC OSX. This free PBX system offers you all features and functionality provided by any hardware based proprietary PABX. It also supports VoIP, SIP etc. It has built in IVR ( Interactive Voice Response) facilities as it is a PC based PBX system.
Now a days, all CTI boards have support for Asterisk, including CT giant Dialogic.

2. FreeSWITCH

FreeSWITCH is an open source telephony platform designed to facilitate the creation of voice and chat driven products scaling from a soft-phone up to a soft-switch. It can be used as a simple switching engine, a PBX, a media gateway or a media server to host IVR applications using simple scripts or XML to control the callflow.
So, it is essentially a software based PBX with many modern CTI based applications built in. It can be installed in Windows, Linux, Sun Solaris. It uses many free code or code snippets from other free PBX software like Asterisk, openSER etc.

3. OpenPBX

OpenPBX a full function, web enabled PBX application, suitable for small office installations and can scale to large call centres. Features include a web based user and management GUI, unlimited Voicemail, Hierarchical Auto-Attendant, Automatic Call Distribution ACD, Least Call Routing (LCR), Music on Hold (MOH), Call Display Records (CDR), unlimited huntgroups, call transfer, call parking, call baring. It has the ability to offer 3 way call conferencing and by leveraging the power of the desktop it offers voice to email, click to dial and transfer of calls.

OpenPBX is supported by VoiceTronix.

Apart three above, there are few more open source, software based PBX systems which could be modified or customised to use as complete PBX solutions.

Though, Asterisk and FreeSwitch have many die-hard fans as well as users, I still feel, traditional hardware based PBX systems are more suitable as they have support and service back up. Also, many offices or corporates may not want a system which follows the principle of do-it-yourself. Instead of hiring few people to continuously develop or customise a PBX systems, many would prefer to buy a hardware PBX with enough features.

But I hope, one day open source PBX would grab their market share just like Linux.