A summary of the argument is as follows:
If the world were to end soon, then the population of the world as it now stands is approx one tenth of the total population of humans who have ever lived. However, if the world were not to end tomorrow, our current population is probably around one billionth of the population of humans who will ever live. In choosing between these two scenarios, the fact that I am now living to ask the question allows me to calculate the probability that the world will end soon, and this turns out to be of the order of 99.999999% 
The first person to discover the result was astrophysicist Brandon Carter, and it was later published by philosopher John Leslie, a prolific author on the subject. Versions of the Doomsday argument have also been independently discovered by other authors. In recent years, there have been numerous papers trying to refute the argument, and an approximately equal number of papers refuting these refutations.
- Summary of the argument adapted from: Aranyosi,I.A., (2004), The Doomsday Simulation Argument; available from http://philsci-archive.pitt.edu/archive/00001590/
- Leslie, J. , (1996), The End of the World, Routledge
- See for example: Nick Bostrom's summary on http://www.anthropic-principle.com/primer1.html