First World Countries
The United States, the UK, Australia, New Zealand, Japan, and most of western Europe formed the 'First World'. This was essentially NATO and America's allies.
Since the end of the Cold War, the term 'First World Country' has stuck and been used (rather incorrectly) to refer to richer countries. The modern is use is somewhat problematic as some African countries would have been considered 'first world' and some rich European countries would have been considered 'third world'.
Second World Countries
The USSR, China, Mongolia, Vietnam, Cuba, Ethiopia, South Yemen, and most of eastern Europe formed the 'Second World'. These were the USSR's and/or China's allies during the Cold War.
Some of the countries would now be considered 'Third World', although quite a few might also be considered 'First World'.
Third World Countries:
Switzerland (yes, that's right), Finland, Sweden, Ireland, Brazil and much of the Americas and Africa was referred to as the 'Third World'. They didn't align with either the USSR or the US, and had no organised alliance amongst themselves.
As many of these countries in Africa and South America are now poorer countries, the term 'Third World' has been used to describe poorer countries. It's pretty obvious to see that this isn't a good term to use, as Switzerland and other neutral countries were also part of the 'Third World'.
So to answer your question, it's a byproduct of Cold War geography and international relations. It has no use today, and strictly speaking neither do the terms 'First' and 'Third' world.
