Natural pearls are those pearls which are formed in nature, more or less by chance. Cultured pearls, by contrast, are those in which humans take a helping hand. The only difference is that in the one case, the process was begun accidentally; while in the other case, it was begun intentionally.
The technique of producing spherical cultured pearls was developed in Japan, and the culturing of pearls is a major industry today in many parts of Asia, including China, Thailand, and the South Seas of Tahiti. Artificial pearls, in contrast to cultured pearls, are entirely artificial, made largely of glass.

Pearls have been cultured successfully since 1920. In this process a mother-of-pearl bead, from three-quarters to nine-tenths of the diameter of the desired product, is introduced into the tissue of a pearl oyster. Over a period of years the oyster secretes layers of nacre around the bead. As the nacre builds up in layers, it surrounds the irritant and eventually forms a pearl. Cultured pearls are not easily distinguished from genuine pearls except by an expert. 
Modern-day cultured pearls are primarily the result of discoveries made in the late 19th and early 20th centuries by Japanese researchers, most notably Kokichi Mikimoto.  What Mikimoto discovered was a specific technique for inducing the creation of a round pearl within the tissue of an oyster.

This discovery revolutionized the pearl industry, because it allowed pearl farmers to reliably cultivate large numbers of high-quality pearls. In contrast to natural pearls, which have widely varying shapes, sizes, and qualities, and which are difficult to find. The pearls can be grown by the tens of thousands, thereby bringing their cost down to a point where cultured pearls became accessible to large numbers of people around the world.
Cultured pearls can often be distinguished from natural pearls through the use of x-rays, which reveals the inner nucleus of the pearl.