Plato's myth of the cave appears in book VII of The Republic, in a dialogue beween Socrates and Glaucon.
In the chapter, Socrates talks about a cave where people are chained facing a wall opposite the entrance to the cave. People outside the cave carry objects around so that the inhabitants of the cave see the shadows of the objects. Now the people in the cave who only see the shadows think that the shadows are actually the objects themselves. The book goes on to where Socrates (Plato's alter-ego in the book) explains what he means by this story.
This story illustrates Platos idealism. Where the objects that we sense (see, hear, touch, etc.) are but shadows of the real things. For Plato, my old philosophy teacher used to say, the real things exist in a "world of ideas."
This story builds on the question that the pre-socratics posed: are what we see around us real or mere illusions? And this theme recurs many times througout the history of Philosophy. This led to Augustine and Descates to ask: if what we sense are illusions, what truths can we know for certain? (The answer, the existence of the self, the famous I think therefore I am.)
This theme also is familiar to those who know Kant's noumenon (the thing in itself) and phenomenon (the thing as we perceive it to be). Of course, in these modern times, with computers and virtual reality the myth of the cave does not seem far-fetched anymore.
The Matrix is the myth of the cave with special effects.