Neo-Platonism is an ancient school of philosophy beginning in the 3rd century A.D. It was based on the teachings of Plato and Platonists; but it interpreted Plato in many new ways, such that Neo-Platonism was quite different from what Plato taught, though many Neo-Platonists would not admit the distinction.
Neo-Platonism began with the Alexandrian philosopher Plotinus, though Plotinus claimed to have recieved his teachings from Ammonius Saccas, an illiterate dock-worker. His most important work was the Six Enneads, in which he explains his philosophy.
Plotinus taught the existence of an indescribable One, which emananated the rest of the universe as a sequence of lesser beings. Later Neo-Platonic philosophers, especially Iamblichus, added hundreds of intermediate gods and beings as emanations between the One and humanity; but Plotinus' system was much simpler in comparison.
Neo-Platonism was frequently used as a philosophical foundation for paganism, and as a means of defending paganism against Christianity; but many Christians were also influenced by Neo-Platonism. Most important of these was Pseudo-Dionysius the Areopagite, whose work was very influential in the Middle Ages. Augustine was also deeply influenced by Neo-Platonism.