How did the slave trade function?

The transatlantic slave trade was triangular and involved three major steps, linking three continents' economies and resulting in the deportation and sale of about 12.5 million individuals. Ships departed Western Europe (more specifically, from ports like Bristol and Liverpool in Great Britain) bearing manufactured goods that could be traded: tobacco, cloth/textiles, gunpowder, guns, rum, etc.
When the ships arrived in Africa, these goods were swapped for human life at various coastal forts and compounds established by trading companies. The slaves were African individuals who had been captured either during raids by European traders or sold by African and African-European dealers who kidnapped them and marched them to the coast. 
These slaves were then brought on ships to the "New World" via the Middle Passage. This was an arduous trip which resulted in the death of many slaves who were living in unsanitary, cramped conditions. Upon arrival, the surviving slaves were sold to the American colonies or to the West Indies at auctions and forced to work for plantation owners, yielding cash crops. The result of this agricultural labor--sugar, cotton, tobacco, coffee, and rice--was then brought back to Europe by those very same ships.
This horrific and inhumane system began in the early 16th century and continued on into the mid-19th century. 
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