Metaphysics means "above" physics. Although Plato was preceded by other metaphysical philosophers, most directly by Paramenides, I consider him to be THE classical/ancient metaphysical philosopher, meaning his influence was the greatest of the ancient world.
His metaphysics, as the first poster noted, has to do with Ideal Forms. These are the concepts which are the metaphysical ideas which provide the concept from which objects manifest or 'form' themselves. There is the concept of a chair and the real physical chair. The Ideal Form of the chair, which is defined by its function and form is the perfect chair. The actual physical chair is once removed from that Ideal Chair and a painting of the chair is twice removed from that Ideal Chair. This system of thinking is also exemplified by Plato's most famous "Allegory of the Cave."
People are chained, unable to move, facing one wall in a cave. There is a fire behind them and figures passing in front of the fire, also behind them. For them, reality is the shadows on the wall. Plato thought what would happen if a prisoner was freed and able to see that the figures using the fire were more real than the shadows. He further thought, what if the prisoner made it outside the cave to the Sun, the Ideal Reality. He answered this by saying the philosopher could get out and must go back in to try, perhaps futilely, to teach the others to break the bonds of their illusions of reality. The parallels are the Ideal Chair/Sun, physical chair/fire and figures, and painting of a chair/shadows on the wall. This was the beginning of a metaphysics which showed how reality is based on perception and that maybe Actuality, the objective Ideal, might be obtainable. These ideas of reality being based on perception and the concept of Ideal Ideas still circulate in philosophical discussions and even in science where the goal is objective observations: cue the Sun.
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