The term “Neoplatonism” is used to describe various philosophical systems created after Plato’s death that build upon the work of Plato. The people modern scholars call “Neoplatonists” actually used the term “Platonist” (not “neo-”) to describe themselves.
In 20th and 21st century use, the period immediately after Plato’s death is referred to as the “Old Academy”. The Hellenistic period, in which the Platonic academy engaged in sceptical rather than dogmatic philosophy, is referred to as the sceptical or middle academy, a period in which eclecticism (a mingling of various different philosophical schools became common).
Neoplatonism proper starts with Plotinus, who is one of a group of 3rd century philosophers particularly interested in the mystical aspect of Plato’s work, and (in some cases, such as Iamblichus) its relationship to the Pythagorean school. Major ancient Neoplatonists include Plotinus, Porphyry, Iamblichus, and Proclus.
Neoplatonism influenced many ancient Christian thinkers including Augustine, ps.-Dionysus the Areopagite and Boethius, and was a major influence on Christianity on the medieval and early modern periods. Ficino is a very important figure for Renaissance Platonism, and the Cambridge Platonists are crucial for Protestant Platonic thought.