Platonism takes Plato's metaphysics seriously: there are eternal Forms, among them Justice, Beauty and the Good, more real than their changing physical instances, reached by disciplined thought and not by the senses. Knowledge properly so called is knowledge of them.
Its afterlife has been longer and stranger than that of almost any other philosophical position. Middle Platonism, and then Plotinus's Neoplatonism, reworked the system into a hierarchy descending from an ineffable One, and it was in that form that the tradition entered Christianity. Augustine absorbed enormous quantities of it, and through him Platonist structures became load-bearing elements of Western theology: the intelligible order behind appearances, the ascent of the soul, the identification of being with goodness. Ficino's translation of the complete dialogues, printed in 1484, refounded Platonism in Renaissance Florence. The Cambridge Platonists revived it again in the seventeenth century, against Hobbes.
It remains a live position and not a historical curiosity. Mathematical Platonism holds that numbers and sets exist independently of minds and are discovered instead of invented; it is held by a substantial share of working mathematicians, and defended by philosophers who accept nothing else in Plato.
Its politics are the most contested part of the legacy. The Republic's rule by philosopher-guardians, its censorship of poetry and its rigid class structure led Karl Popper to name Plato the first great enemy of the open society. Specialists have largely rejected that reading as anachronistic and deaf to the dialogue's ironies. The question underneath it has not gone away: whether a politics grounded in knowledge of an objective Good can leave room for the people who fail to see it.