Private Branch Exchange, or PBX for short, is a telephone exchange that works with a particular business or office. This differs from the common telephone carrier in that it is not a service for the general public. PBXs can connect the internal telephone lines of private companies, along with connecting to the PSTN (Public switched telephone network) through trunk lines.
In years past, PBX services for smaller companies / offices were notoriously challenging to arrange, and business owners realized that handling their own telephone service was not an ideal scenario. That precipitated the need for hosted PBX systems. Hosted PBX is managed by telephone service providers via the internet. One of the advantages of using this service is that there is no equipment to purchase or maintain. The branch is removed from the private location and shifted to a more central location.