The traditional first, second, and third world comes from French de facto geographer Alfred Sauvy. The model was not economic but political at first. The third world was defined at first as not politically aliened to either the Soviet Union or the United States. Later on those allied to America were defined as "first world" while "second world" were those Communist-states in the USSR's pocket.
Later on as the Cold War ended the world system adopted an economical framework. The capitalistic victors' economies continued to do well. The fall of the Iron Curtain; however, unveiled how the communist system barely tread water. Meanwhile those countries who remained non-aligned varied from second world-level to God awful hell hole.
The worlds model was therefore changed to reflect primarily the economic status and now matches the developed/developing countries model. Economic numbers like GDP, gini coefficent (income disparity), and scale of living are now the main variables.
There is also a relatively new category: the fourth world. The fourth world has been defined in two ways. First a countries who would feel minimal impact if oil supplies were cut off. Somalia may be in this category if it were not for the pirates and Islamic militants who need fuel for their operations. The second definition is that of a nation (people) who are cut off from globalization's benefits. The natives of far eastern Siberia or Hutus in The Congo are the best examples.
There is even a Maoist interpretation of the world's theory. Mao, being ever the Chinese nationalist even in his communism, saw the first world as the Soviet Union and United States keeping the second world, like the United Kingdom or Poland, in the first world's sphere. Mao believed the People's Republic of China and a few others like Albania were the third world who fought for their own way in the world.
Finally there are the microgeographers who do not like to use first, second, third world definitions at a country level. Subnational/district level is the best way to define an area's status. Microgeographers point out that even though the United States is "first world", areas of it are second world like the rust belt and third world like western Indian reservations. Another example is India with its wide range of super-rich and super-poor.