Western world has meant various things at various times. In the earliest history of Europe, it referred to Ancient Greece and the Aegean. During the Roman Empire, it meant the Western Roman Empire (covering the area from Croatia to Britain). At other times, it has meant Western Europe or Europe or Christendom. During the Cold War, it sometimes meant the democratic countries or those allied with the various NATO powers.
In this day and age, it often refers to the places that have a European cultural heritage, called Western civilization. This European heritage itself is a combination of Judeo-Christian ethic, classical Greco-Roman thinking and the cultural practices of the "barbarian" peoples of Northern Europe. Under the strictest definition, it would be the United States, Canada, Western Europe, Australia, and New Zealand.