A few different instances resulted in African slaves being brought over to the New World, including the Caribbean, in the 1500s. Most of the events that resulted in slaves being brought to the Caribbean only brought a few slaves at a time.
The first record of African slaves in the Caribbean occurred during Diego Cólon's time as the governor of the Spanish empire in the Caribbean. Cólon believed that Native American slaves were not hard workers, therefore he started the process of petitioning for African slaves to be shipped there. King Ferdinand of Spain allowed for the shipping of 50 African slaves to be sent to the Caribbean in 1510.
Throughout the course of the rest of the century, slaves were sent to places like the West Indies and Jamaica, to work in tobacco and sugar fields, but the slave trade did not increase to large numbers before 1600 because several countries, including France and even Spain, fiercely debated the legality and ethics of slave trade.