What is the conventional interpretation of Allegory of the Cave in Plato's Republic?
In Plato's Republic, according to the conventional interpretation, the allegory of the cave is an extended metaphor used by the Socratic character to explicate what is usually called "the middle theory of forms", a theory about the nature of reality and knowledge held by Plato in his middle period, but not fully formed in his early dialogues and rejected in his later dialogues. It is extremely unlikely that such a theory was held by the historical Socrates, even though Plato uses Socrates as a mouthpiece to articulate it.
The 'forms" or "ideas" are non-corporeal ideal realities which the objects of everyday life imitate. Just as the people in the cave only see vague monochromatic two dimensional shadows and understand little of the actual world outside, so our ordinary experience is only that of vague and misleading imitations, or shadows, as it were, of the transcendent forms.