Actually, the "African tribes" didn't simply have countries but had empires like Mali, Songhai, Benin & Soninke that were in existence well before there was any real political unity in Europe.
But, yes, the slave trade in Africa reached back at least to 700 BCE and in terms of human history has been part of us since before the Stone Age, I imagine.
Were there "African" slave ships hauling African-slaves to the Americas and the Caribbean? No, those were Portuguese, Spanish, English, French, Dutch barques. The slave-trade running to Asia from Africa was facilitated by Arab traders.
Those African empires were not primarily sea-faring at all. Thought they did a remarkable slave-trade in land-routes across the Sahara, up to Morocco and down to Benin and "the Guinea Coast."
Slavery was and has been endemic to human populations. However, it's rise to the level of a "world leader," economically, was made by European traders.
If you read-up a bit on "indentured servitude" you'll discover that black slavery was not the only variety of slavery practiced in the New World. Whites were in fact subjected to it as well. The major difference was that whites could and did escape by running away and our skin color didn't mark us out as being "slaves," or in fact "indentured servants."
Nichole