There are several different ways of looking at the slave trade, but one of the most common descriptions of it is the triangular trade. European merchants brought goods to Africa to trade with the Africans who controlled the slave trade in Africa. They traded these goods for slaves which they then brought to the Americas and traded for cotton, sugar, rum and other goods from the Americas that were generally produced with slave labor. The merchants then returned to Europe bearing the goods from the Americas.
The Africans generally controlled the actual slave market, bringing in captives from tribal warfare as well as from traders who specailzed in simply capturing people for slavery.
The Europeans dominated the other aspects of the trade because they dominated the shipping trade.