The Free World is a term that was used during the era of the Cold War. It was often used by non-communist nations to describe themselves. It was used to describe the greater personal freedom that the people of non-communist countries (such as the United States and Western Europe) had compared to the Soviet Union and its East European allies.
On the other hand, many of the non-communist countries that were allied with the "Free World" during the Cold War repressed people and sometimes had dictators as their leaders. This included countries in South America, Asia and Africa.
The President of the United States was often called the "leader of the Free World". This was very common especially in the United States itself. The President of the U.S. is sometimes still called this.
Neil Young sang a song called "(Keep on) Rockin' in the Free World".