There are many things keeping the third world poor.
Population increase is one, but then again its always been the way for "poor and uneducated" people to secure ones old age. The problem here, is that we have gotten better to save babies and children with modern medicine, even in the 3rd world. In the old days child mortality was huge, so they needed more babies to keep populations up. This mentality is still in place in many areas of the planet, especially where women have almost no rights over their own bodies. Add to that we have gotten better in food production and helping famine areas around the world.
But take China, better medicine, longer lives but same production of babies and lack of disasters, wars and famine = 1 billion people plus. Chinese have always been high baby producers, but that was because of cultural aspects, like children taking care of their elders. To make this viable they needed children, children that lived.
Another huge factor in India and even China, was the wish for male children. Now days saddly they just abort female fetuses because they can see the sex via modern medicine. This saddly gives a wacked out balance between women and men.. in India its getting closer to 40% women and 60% men... and that is scary. Same thing is happening in China.
However saying that, this problem might just solve it self in Africa at least, in the most horrible way.. AIDS. With some nations populations over 30 to 40% infected, its only a matter of time. But that will not solve the India and China problems, unless AIDs spreads like wildfire in those countries (and we dont know as both goverments are rather... lets say protective of the facts).
On the political front, the 3rd world was left by its former colonial masters in shambles. Africa with the exception of South Africa and some what Egypt, was left to themselvs, without localy educated technocrats or goverment. With such a power vacum, many saw fit to grab power and well thats been the history of Africa for the last 60 years.. war and dictators. This also had impacts on the infrastructure, that was crappy at best. South America faired a bit better, as many former colonial powers citizens stayed behind to run things, plus goverment and educational institutions were created there. However the political instability there during the last part of the 20th century hurt them quite a bit.
On the economic front, the 3rd world is only good for 1 thing.. cheap labour. However the lack of educated workers is a problem, with the execption of China and India. Africa is hopeless and totaly unusable for anything frankly and hence the lack of investment there. However if the political and educational, and infrastructure improves there, then who knows.
Then there is of course the first world who do keep 3rd world countries "down" via their economic policies and even political policies, but thats out of self interest.