Slave trade in Africa by the Muslims was already in place since the Middle Ages. However, transatlantic slave trade can be traced precisely to the trial in deffense of the Indians by Father Las Casas. He was fighting to abolish the explotation and genocide of Amerindians, and to get that goal he had to accept the Spanish settlers of Cuba to replace Indian with African labour.
That was the start of the atlantic slave trade.
Between the British, perhaps the first slave traders were corsairs like Francis Drake, that make money transporting and exploting slaves in the Caribbean. The British crown invested in those "business" as well by buying shares.
Britain and Portugal were the largest slave trades of all. The Caribbean imported more slaves that all the other regions together. Few of them survived more than some years of explotation and didn't left descendency.
Pinguin