The answer to this question covers such a vast expanse of time, territory, and peoples, that in this particular forum I will only be able to give you the broad strokes—however, I will also gladly provide you with some references for further research.
In the first place, the slave trade caused the involuntary export of anywhere from 8 million to 20 million human beings, in a period of time stretching from the 1400s, which were the earliest days of the Atlantic slave trade, all the way up to the mid-nineteenth century—and this range of numbers represents a conservative estimate. To get an idea of the scope of this number of human lives, the population of the United States as a whole at the outset of the American Civil War was in the neighborhood of 31 million people. To say that the removal of tens of millions of men, women, and children would have a deleterious effect upon the social development is a gross understatement. In many sub-Saharan African nations, it is hypothesized that the extraction of such huge numbers of people had such a deleterious effect upon the economy of those nations, that they never recovered from it; it is speculated that because of this, the course of the development of those nations was irrevocably transformed.
Secondly, the social development of African as a continent of nations, the social development within those nations, and the perception of African and persons of African descent, was indelibly changed by the slave trade. The precise effects are a matter of some debate, as are the precise numbers of human beings taken into bondage in Africa. However, Walter Rodney, who was a prominent Guyanese historian and prolific commentator on the effects of the African slave trade, stated that the population of Africa stagnated for centuries following the inception of the slave trade, while the populations of Europe and the Americas continued to increase.
Politically, it has been speculated that the slave trade caused factionalism within ethnic groups and thus irrevocably changed the structure of power in many African nations. Historical consequences of war between factions, such as enslaving one’s enemies, now became profitable as these enemies could then be sold to European slavers. This may have further destabilized the political structure in many areas, because it may have provided an impetus for conflict. Another tragic consequence is the perception of Africa and African peoples by outside nations and peoples. Africa, a continent vast in territory and ancient in history, with rich cultures and civilizations, artistic and technological history, became a source of human slaves who were seen by outsiders and by slave traders and those societies dependent on slave labor, such as the American South, as being not even human. Laws codified this racism: the Three-Fifths Compromise of 1787 during the United States Constitutional Convention was designed to level the number of representatives, and thus, the amount of influence, that Southern states had. For the purposes of determining numbers of representatives, Southern delegates preferred that their huge numbers of slaves be counted as population, which would have provided the South with more representatives. However, slaves were property and not even thought to be human, and therefore had no voting rights. The compromise was that only three-fifths of the slaves could be counted for population figuring purposes and thus for the distribution of representatives of the Southern states. Events such as this led to a tragic result: once enshrined in law, the fight for equality for black Americans, and the ripple effects of institutionalized racism, with all of its attendant human brutality and misery, is suffered in the United States to this day.
As I said, the question of slavery and its effects is hundreds of years old, and affected millions of people of many very different cultures and nations in the continent of Africa. For more information, I encourage you to look further into the works of Walter Rodney, in particular, his book How Europe Underdeveloped Africa; and also into the delightful Ebony and Ivy: Race, Slavery, and the Troubled History of America’s Universities by Craig Steven Wilder, among many, many other books.
I also encourage you to explore the resources of the brilliant Charles H. Wright Museum of African American History in Detroit, Michigan, which can be found online at thewright.org.
I hope this helps.