The present day borders of the countries in Africa are a colonial legacy, that is, they were drawn with the specific purpose of exploiting the resource potential and assuring colonial control as opposed to what would best serve the interests of the people in those countries for future development. Africa is suffering the long term effects of this colonial legacy, coupled with the aftereffects of the Cold War and its massive infusion of weapons into African militaries and society.
The colonial powers, especially Britain, were sure to draw borders around resources they could use, and to do so in such a way that minority tribes who had long been oppressed by majority tribes would end up in the same country together. Britain would then empower the minority tribe with education, government jobs and privileges. In return, the minority tribes would remain loyal to the British and assure the flow of resources was unimpeded. This was called indirect rule, and while it worked well from a colonial standpoint, it built up long term resentment from an African one, and assured that wars and civil strife would be the order of the day long after the empires had left.
Post-African independence, that is, starting around 1961, both the Soviet Union and the United States courted African leaders with foreign economic and military aid. This served to reinforce dictatorships and oppression as those same leaders played the two superpowers against each other and lined their pockets at the same time. As the Cold War eventually came to a close, the same enemy tribes stuck with colonial borders were still armed to the teeth.
I believe it is these two legacies which are directly responsible for the lack of economic development in African nations, and the ongoing human tragedy in most of them today.
Just a quick addition concerning how colonialism contributed to the economic underdevelopment of Africa. The borders of African countries today were drawn by European colonial powers long ago. When these borders were drawn, there was no consideration concerning ethnic and cultural similarities and differences. So today there are African nations with populations that contain different ethnic and tribal groups that have long histories of conflict. These ethnic conflicts continue today, causing many African nations to become politically unstable and difficult to govern. The political chaos caused by these ethnic differences has helped contribute to the economic chaos and underdevelopment experienced by these African countries.
I think that colonialism contributed in a variety of ways to the economic underdevelopment of many nations and continents. Africa was one very powerful case, though. The consistent and thorough plundering of indigenous resources for the parent nation's gain was one way in which the underdevelopment of Africa was present. Once the nations in Africa gained independence, they discovered that there was very little upon which to build to establish economic footing and a sense of economic autonomy. There came a point where all of the parent nations had done a very good job of essentially robbing many of the nations in Africa of their economic livelihood based on natural resources. Another way in which colonialism "did a number" on nations in Africa was in the lack of establishing a consistent and effective economic infrastructure. For over 200 years, parent nations from Europe had controlled African nations and had not allowed Africans to control their own economic interests. Kept in subjugation in both physical and political terms, when these nations were given independence, many of them realized that fighting for freedom and economically managing it were two different realms entirely. The chaos that resulted from this realization helped to further the economic underdevelopment of Africa. Along these lines, the lack of a mandated and successful economic infrastructure plan of development was painfully evident in these nations. When independence was gained, the parent nations simply left, having taken what they wanted and doing little to ensure the new nation's future from an economic standpoint. Many of the parent nations did little to economically support the new and fledgling nations in Africa. Essentially, one had the situation where one new nation was in economic competition with nations that had been dominant from a material standpoint for over 200 years. It was no contest, and helped to contribute the economic underdevelopment of Africa.