Of course, there's no single, simple answer. Different countries are poor for different reasons. Poor countries have become rich, and rich countries have become poor, often due to circumstances beyond their control. Sometimes it's a combination; Chile was hit by a perfect storm in the early 1970s as the price of their chief export, copper, fell in world markets, and the failed socialist policies of the Allende government sent the economy into a whirlwind of hyperinflation and GDP contraction. Only now, after suffering through years of dictatorship and free market experiments has Chile's economy now become the most dynamic of Latin America.
But for many countries, there are no such dynamics. These remain desperately poor, and have little hope of digging out of the hole they are in. They have an insufficient capital base for needed investment, and through increasing population or deteriorating environmental conditions, what little they have actually decreases on a per capita basis.
Many countries in Sub-Saharan Africa fit this bill, whereas other countries afflicted with extreme poverty such as Bangladesh or Cambodia are at least on path to development, with economic growth that is self-sustaining and benefiting from positive feedback.
So what's wrong with Africa?
Look at the map, and the first thing you should notice is the high number of landlocked countries, cut off from easy access to ports and international shipping. Even if individual villages are capable of surplus agricultural production, there is no easy way to market these goods to other countries. Even marketing to neighboring villages can be a huge challenge, due to insufficient road networks. The infrastructure does not exist, and since these places are so poor, the governments cannot finance its construction without foreign aid and investment.
Another factor is public health. Africa is by far the most diseased continent. Not only is the incidence of malaria higher than anywhere else, but the particular strains native to Africa are much more deadly. AIDS is also a huge problem, and like malaria, it afflicts Africa like nowhere else, with millions already dead, and millions more orphaned or debilitated because of the disease. Again, fighting sickness in Africa hits a wall due to insufficient public investment and infrastructure. In the case of malaria, incidence of the disease can be reduced dramatically through the use of mosquito nets. AIDS can be combated through education and basic safety measures. But the resources are not there.
It's a vicious circle, as an unhealthy workforce is also an unproductive one, and an unproductive workforce cannot finance desperately needed public investment in sanitation, education, health care, and improved transportation infrastructure.
The only way to help these countries out of this rut and on the path to development is for the rich countries to open their wallets, and get serious about ending poverty. Because it clearly has not happened yet. Canada's foreign aid budget amounted to less than 0.3% of GDP last year. The US is worse (though improved from previous years), at only 0.2% of GDP. The most generous donor countries are the Scandinavian nations, who give very close to 1% of GDP. The truth is that if all these donor countries pledged 1% of GDP for the next couple decades, extreme poverty could be virtually eliminated throughout the world. The UN Millennium Development Goals were intended to halve poverty by 2015, but it doesn't look like they will be met.
Unfortunately, the political will simply does not exist, partly shaped by mistaken assumptions about why the Third World is poor. It is not because of "corruption". It is certainly not because people in these countries are lazy. Racist ideas about the destinies of nations become absurd when you examine each country in context, even more laughable when taking into account the decades of colonial exploitation many of them endured.
It's also remarkable how people in the US and other rich countries routinely overestimate their government's aid budgets many times over. Many Americans mistakenly identify foreign aid as one of the biggest pieces of the federal budget, when it is nothing of the sort. Far less than 1% of the US federal budget goes to anything resembling foreign aid (there are similar misconceptions about welfare). These misperceptions will have to be overcome before extreme poverty can be eliminated.
For further reading, I recommend Jeffrey Sachs' The End of Poverty.