The analogy of the line has to do with the theory of knowledge. Plato recognized that knowledge is better than opinion. If Euthyphro was to know what piety is, he must know it through the form, which can only be thought and not sensed. Thus knowledge belongs to an invisible, intangible, insensible world of the intellect, while of the visible, tangible, sensible world we have only opinion. The intelligible world is more real and true than the sensible world, as well as being more distinct.
Suppose we say in the abstract that there is some proportion of reality, truth and distinctness between the invisible and visible worlds. This can be represented on a line. (You can suppose the ratio be whatever you like, say 3:1).
Now Plato says that within each realm there is a further division. In the realm of the visible, there are real objects and their images (shadows, etc.). The images give us the lowest grade of belief, mere conjecture. If I see a shadow of an object, I get very little information about what specific object it is. Plato lays it down that the proportion of truth, reality and distinctness holding between the object and the image is the same as that holding between the intelligible and sensible worlds (e.g., 3:1).
Similarly, there is a division within the intelligible realm, between the forms themselves and images of the forms. Knowledge of the forms themselves through reason is the highest kind of knowledge, while knowledge of the images of the forms through their images through the understanding is a lower form. (Again, the ratio would be 3:1).
This identification may perhaps be understood in this way. Our opinions about the objects of the world are formed through the use of the senses, by observation. We can observe that things tend to go together all the time, and thus form the opinion that those things belong together. If Euthyphro had the right information about the preferences of the gods, he could observe that certain acts are pleasing to all of them. But he has not explained anything. He is left with mere opinion.
We might try to understand objects of the visible world by using our understanding. We can make assumptions and show what follows from them. The use of these assumptions can enable us to generate laws which explain why things go together the way they do. For example, Newton assumed that bodies in motion tend to stay in motion, and bodies at rest tend to stay at rest, unless some outside agency acts on them. This assumption about inertia helped him generate further principles about motion, but it is not itself proved. It is an unexamined assumption, in Plato's terms. This method of proceeding is not the best way possible. One must instead start with forms and use them in explaining other things
. The cave analogy is in many respects similar to that of the line. It distinguishes between the most true, real, and most distinct (in this case, it is compared to the world outside the cave) and the least (the shadows in the cave and higher than them the objects in the cave casting the shadows when illuminated by fire within the cave).
The difference between the analogies is that the cave analogy is more vivid in its depiction of the sensible and intelligible realms, and that it illustrates the problems of coming to know through the forms. Each step in our progress, from conjecture, to opinion, to knowledge, has its difficulties. The images on the wall of the cave are easily mistaken for the real if they are all one can experience. When one breaks free and looks toward the fire, the objects casting the shadow are now mistaken for the truly real, and the light of the fire is painful and dazzling. This effect of bewilderment is even more intense outside the cave. Here, however, one has reached the real at last. Finally, if a person trained by the state reaches this higher form, he has the responsibility to govern. The philosopher-king knows the good itself, and hence knows what is good to do.
A last point about the forms. They are what gives us knowledge, but they are also what gives things their reality. The sun casts light upon the earth, allowing us to see what is there, and it also supplies the energy through which things grow and prosper. So the form of the good gives to the sensible world the reality it has. Later philosophers in the early days of Christianity were to adapt this image of the sun into a thought of God as the source of all reality and knowledge.