Plato, answered from the texts and cited to the page.
The sun analogy is the argument, set out in Republic VI, that the Form of the Good stands to the intelligible realm as the sun stands to the visible realm. Let me say what I mean by that. Consider sight. Of all the senses, it alone requires a third thing besides the organ and its object: without light, the eye cannot see, and colors cannot be seen, however present both may be.1
That third thing is the sun — not itself sight, not itself color, but the cause and sovereign of both.2 The sun is not sight, yet it is the cause of sight itself, and it is seen by sight; and beyond giving things the power to be seen, it gives them coming to be, growth, and nourishment, though it is not itself coming to be.3 Now carry the analogy upward.
What the sun is in the visible realm, the Form of the Good is in the intelligible. It gives to the things known the power to be known, and to the knower the power to know; it is the cause of knowledge and truth, yet it is something other and more beautiful than either.4 Knowledge and truth are beautiful, but the Good exceeds them, as the sun exceeds the light it produces.
When the soul turns toward what is illuminated by truth and being, it understands and knows; when it turns toward what is mixed with coming-to-be and passing-away, it merely opines, shifts its opinions, and seems bereft of understanding — just as eyes see clearly in sunlight and go dim in darkness.5 The divided line then extends the image.6 The visible realm divides into images (shadows, reflections) and their originals; the intelligible realm divides into the section where the soul reasons from hypotheses toward conclusions, using visible figures as images, and the section where it ascends from hypothesis to a first principle that is not a hypothesis at all, using the Forms themselves.7
Geometry is the intermediate case: the student hypothesizes the odd and even, the various figures, without giving any account of them, and proceeds downward to conclusions — his thought is directed not to the drawn figures but to the square itself, the diagonal itself.8 The philosopher goes further, using no images, ascending to the unhypothetical first principle.
And the cave, which follows in Republic VII, is the image of the whole ascent: the visible realm is the prison, the fire inside it is the sun's power, and the upward journey out is the soul's journey to the intelligible realm, where the Form of the Good is the last thing seen and the hardest — yet once seen, it is recognized as the cause of all that is correct and beautiful in anything, the source of truth and understanding in the intelligible realm, of light and its source in the visible.9 Anyone who is to act sensibly in private or in public must see it.
Sight may be present in the eyes, and the one who has it may try to use it, and colors may be present in things, but unless a third kind of thing is present, which is naturally adapted for this very purpose, you know that sight will see nothing, and the colors will remain unseen.Complete Works, pp. 1157–1158
The sun is not sight, but isn't it the cause of sight itself and seen by it? That's right. Let's say, then, that this is what I called the offspring of the good, which the good begot as its analogue.Complete Works, pp. 1157–1158
the sun not only provides visible things with the power to be seen but also with coming to be, growth, and nourishment, although it is not itself coming to be.Complete Works, pp. 1158–1160
what gives truth to the things known and the power to know to the knower is the form of the good. And though it is the cause of knowledge and truth, it is also an object of knowledge. Both knowledge and truth are beautiful things, but the good is other and more beautiful than they.Complete Works, pp. 1158–1160
When it focuses on something illuminated by truth and what is, it understands, knows, and apparently possesses understanding, but when it focuses on what is mixed with obscurity, on what comes to be and passes away, it opines and is dimmed, changes its opinions this way and that, and seems bereft of understanding.Complete Works, pp. 1157–1158
there are these two things, one sovereign of the intelligible kind and place, the other of the visible... you have two kinds of thing, visible and intelligible. Right. It is like a line divided into two unequal sections.Complete Works, pp. 1158–1160
In one subsection, the soul, using as images the things that were imitated before, is forced to investigate from hypotheses, proceeding not to a first principle but to a conclusion. In the other subsection, however, it makes its way to a first principle that is not a hypothesis, proceeding from a hypothesis but without the images used in the previous subsection, using forms themselves and making its investigation through them.Complete Works, p. 1160
students of geometry, calculation, and the like hypothesize the odd and the even, the various figures, the three kinds of angles, and other things akin to these in each of their investigations, as if they knew them... their thought isn't directed to them but to those other things that they are like.Complete Works, p. 1160
In the knowable realm, the form of the good is the last thing to be seen, and it is reached only with difficulty. Once one has seen it, however, one must conclude that it is the cause of all that is correct and beautiful in anything, that it produces both light and its source in the visible realm, and that in the intelligible realm it controls and provides truth and understanding, so that anyone who is to act sensibly in private or public must see it.Complete Works, p. 1164