The "Free World" was a term much in use during the Cold War which has recently fallen into disuse. Nowadays you're more likely to hear about "the West" or "all civilized nations". These are all very interesting designations. As sekicho's eclectic list of the nations included in the "Free World" shows, the countries included don't really seem to share much in common apart from the fact they were aligned with the United States of America in the Cold War. It's also clear that as the "Free World" evolved into "the West" or "all civilized nations", it changed: Russia is a part of the latter two but was the arch-enemy of the former. What do these designations tell us about world politics?
Some would argue they don't really tell us much beyond shedding light on the prejudices of the U.S. The concept of the "Free World" was articulated by the U.S. after World War II as the superpower took on the role of "Leader of the Free World". There was a widespread feeling in the U.S. that as the victor in the war, the U.S. inherited the responsibility of leading the free nations of the globe into a better world. Like most identities, the concept of "the free nations of the world" became particularly salient when faced with its enemy: the Communist bloc.
America felt it had a responsibility to defend the Free World from Soviet aggression. Communism, looked at from an American perspective, had two salient features: firstly, it was antithetical to American values, and secondly, it was inherently aggressive. Totalitarianism, of which Communism is a subset, was clearly incompatible with the American values of free markets and free politics, and so could not be permitted to wage aggressive war (military or ideological) against the Free World. This essentially Manichaen idea of Communism vs. anything non-Communist came to dominate the American view.
Herein lies the contradictory nature of the "Free World". Many nations within it were not free in the American sense as they suffered human rights abuses, were not democracies, and did not have an active free press and civil society. Simply being anti-Communist was enough to buy you a ticket to the Free World. Communism is so deeply antithetical to American values that many things could be justified in fighting it. Dictatorships were far from ideal, and American support of them could be viewed as hypocritical. Yet hypocrisy is the price of priorities, which do not allow one to address all the evils of the world at once.
Totalitarianism was viewed as so crushing of a nation that the damage wrought by it could never be undone, whereas authoritarian societies were believed to be capable of reform, especially once the Communist threat had passed. One important aspect of the Free World was that it was believed to be indivisible - a threat to freedom anywhere was a threat to freedom everywhere. In other words, any act of Communist subversion against any non-Communist society was seen as a global problem. Having taken on the role of chief global anti-Communist, the U.S. felt obliged to intervene on all Continents to stem the Communist tide. To allow a victory for the Communists anywhere would be seen as abrogating American leadership and only encouraging further enchroachments.
The Free World was hence a very useful concept for America during the Cold War, as it allowed it to form a comity of nations with shared goals and shared identity which they sought to protect. What brought these diverse nations together was their shared insecurities about Communism and its global pretentions to power. Although they had different characteristics and different levels of domestic freedom, they shared a basic opposition to totalitarian Communism. Many which weren't liberal democracies went on to become ones.
The term "Free World" has nowadays fallen out of use, for a number of reasons. Firstly, it carries a lot of the baggage discussed above. Being initially designed to mean anti-Communist, it became irrelevant with the fall of the USSR and the final end of Communism as a credible ideology.
It has been replaced by two terms: "the West" and "all civilized nations". The West is usually understood to refer to most of Europe, North America, Australia and New Zealand, and sometimes Japan. The future of this concept is nebulous - some nations like the Ukraine seem to want to become a part of it, while others such as Japan and Russia seem to want to move away from it. The Iraq war anyway proved that as a basis for common action and mutual support, "the West" is not so useful a category anymore.
"All civilized nations" is a term almost always used to refer to the most common global security anxiety of today, terrorism. The subtext is that civilized nations eschew violence as a means of dealing with their problems except when they have no choice, whereas terrorists see it as their raison d'etre. "All civilized nations" can include a much wider group of countries than the Free World, as it includes basically any opponent of terrorism by non-state actors. Those excluded from "all civilized nations" are basically the state sponsors of terror such as Iran, Saddam Hussein's Iraq, etc. Libya was able to pass into this category by its renunciation of terrorism. This term clearly bears the closest conceptual relation to the Free World, whereas in geographical terms it is very different.