One of the earliest uses of the term free world as a politically significant term occurs in Frank Capra's World War II propaganda film series Why We Fight. In the first film of that series, the "free world" is dramatically contrasted with the "slave world," in the words of the narration. The film depicts the free world as the western hemisphere, lead by the United States and western Europe, and the slave world as the eastern hemisphere, dominated by Nazi Germany and the Japanese Empire. The films portray the Soviet Union as an ally of the free world, which it was, from the perspective of nations fighting Germany in World War II.
Richard Stallman and the free software movement use the term Free World to describe the set of software, people, commercial/non-profit companies, that adhere to the 4 basic freedoms of free software.
So they for example speak of the Free World (note the capital letters) when they refer to a GNU/Linux distribution composed only of free software and they say that for example some free application that use non-free libraries can't be run in a Free World, since in a Free World these non-free libraries don't exist. Debian GNU/Linux (the main distribution of Debian) is a self-sufficient system composed of Free Software, but since the Debian Project also distributes some non-free software, it's not touted by RMS et al.
In a broader sense, lawyers for example are said to live in a Free World, because they have the freedom to study, modify and freely redistribute legal knowledge, decisions and strategies.