Black cultured pearls
Black South Sea cultured pearls, also called Tahitian pearls come from the Pinctada Margaritifera pearl oyster. This oyster is also known as the black-lip pearl oyster. First cultivation of Pinctada margaritifera pearls was successfully carried out in French Polynesia in 1961. For an overview of pearl farming in French Polynesia see this 2011 video.
Though termed black, these pearls present an incredibly wide array of iridescent colours that cover the entire spectrum of light. The dark colour of these pearls is natural and due to organic pigments. They can be found in a wide variety of shapes and with sizes of 6-20mm.
Two other species that fall under the category of black cultured pearls are pearls formed in Pinctada mazatlanica and Pteria sterna oysters. These are cultivated in Mexico, and are often called Cortez pearls.