Malcom X was known to say to African Americans that their history “did not begin in chains.” What is the history of Africa prior to European involvement?

For this I'll use Chapter 9 of McKay et. al's "A History of World Societies", the 8th edition as my primary source. It ain't the most detailed of books, but if you want a general idea I recommend it.

Also, I'm going to narrow down the question a lot: I'm concentrating on West Africa, the area where most African-Americans have their ancestry. I'm also going to concentrate on the slave trade, since it relates to Malcolm X's quote.

Africa's history is very multifaceted - like Europe, it depends on where and when you look. To begin with Malcolm X's comment: he is right in one sense and wrong in another. Yes, the African continent had several notable states and empires (that I'll detail below), but they were no strangers to the slave trade either before the Atlantic slave trade that Malcolm X probably refers to.

The magnitude of the Trans-Saharan slave trade (connecting West Africa with the Mediterranean), in fact, was a big business involving all partners: African as well as Muslim and Christian states, as the slave trade was second only to gold in terms of profitability. It peaked in the period between 900-1100, with nearly 9000 slaves traded anually. It shrunk by roughly half by 1400-1500, but still remained a major trade. [1]

Race had very little to do with the trade: wealthy Muslims in Cordoba and Alexandria owned Christian as well as African slaves. Likewise, West African lords and kings not only sold poor Africans to slavery - they also owned white slaves from as varied backgrounds as Britain, the Balkans and Turkey. [2]

The slave trade is important to African history not only because of the usually-cited circumstances, but also because West African urban centers grew as the trade grew. African rich families congregated in the border zones between the Sahel and the Sahara. By being middlemen between the miners in the south and the Muslim merchants to the north, they could get rich quickly. That's how cities like Timbuktu, Jenne and Gao grew, and other cities likewise became important in the slave trade and other trades. Also, Islam spread to West Africa via the trade, so the lords and kings alongside the population became mostly Muslim quickly.

As the cities grew, empires and kingdoms themselves emerged in West Africa: Ghana (900 - 1100) and Mali (1200-1450) were two of the most prominent, although the Dahomey and the Songhai were also important West African policies. In other parts of Africa, although I'm sidetracking, we have Ethiopia and Great Zimbabwe as prominent civilizations around the same time.

How powerful were the West African empires in their prime? Well, the famous Hajj that Mansa Musa (sultan of Mali) made is a prime example, but they were respected by outsiders as well. Ibn Battuta, a traveler visiting Musa's successor, remarks (excerpt):

"the Negroes possess some admirable qualities. They are seldom unjust, and have a greater abhorrence of injustice than any other people. Their sultan shows no mercy who is guilty of the least act of it. There is complete security in their country."

This could be pretty surprising to someone used to the "Darkest Africa" trope today: it indicates that Mali's society was a relatively secure one with a functioning justice system. In fact, Ibn Battusa further remarks that "They do not confiscate the property of any white man who dies in their country, even if it be uncounted wealth. On the contrary, they give it into the charge of some trustworthy person among the whites."

Now, I should end on an important note: by the time the Atlantic slave trade had started, the principal West African states had lost at least some of their former power. No state had managed to (re)gain unitary and solid control over Western Africa by the time the Atlantic trade began. But the African lords were fine still, even if they were not united - as they now had an extra avenue for selling slaves - until the mid-19th century, when colonialism took hold and slavery lost a lot of its former importance.

So, the answer to Malcolm X is "Yes... and no.". West African states were powerful, but not all of the population were free either. This was not something limited to West Africa - it was normal everywhere at some point, really.

[1] R.A. Austen: "The Trans-Saharan Slave Trade: A tentative census" in "The Uncommon Market: Essays in the Economic History of the Atlantic Slave Trade (Gemery / Hogendorn: Academic Press 1979)

[2] R. N. July: "Precolonial African Economic and Social History" (Scribner, 1975) pp. 124-129

Edit: Too many edits to count.