There is, of course, the historical fact that Plato was Socrates' student. This means that they were two different people; but since their philosophies are so difficult to distinguish, and since all of Plato's philosophy is written as dialogue -with Socrates as the protagonist-, I shall use the two names almost interchangeably.
Plato wrote about many aspects of philosophy. The point that is central to understanding the way he thought, though, is his idea of forms.
Plato believed that the world as we know it is not the Real world. As he saw it, there is another world, one that is unintelligible to our senses. The only way we can come to know this Real world is through Reason. This greater world is the world of things as they truly are; the world we live in is imperfect, and the things that we think we know, we really don't know at all, because knowledge for Plato means understanding the Forms of things.
The form of a thing is what it really is. Things in this world 'participate' in their forms, but are not their forms.
Let me illustrate the point. I am, at this moment, sitting in a chair. But what exactly is it about the thing I am sitting on that makes it a chair, rather than something else? Is it because it has legs? Many things have legs, and are not chairs; and many things that do not have legs, are yet chairs. The same applies for all properties that chairs have (or at least all properties that I can think of). So what is it that makes it a chair? The same question applies to all (or most) things that exist in our world. The answer, according to Plato, is that we understand the Forms of things. I understand that the thing I am sitting on participates in what is the Form of 'chair'.
An argument that Plato gives in support of his view is the 'argument from equality', as I like to call it. I don't remember if this argument comes out in the Meno, Euthyphro, Protagoras, or something else entirely. But here it is:
This is only a rough approximation of what Plato says, but it is exactly the same idea.
Plato believes that there are three parts of the soul. The greatest of these, the one that ought to be in control, is the rational part. The second part of the soul is the passionate part. The rational part, in conjunction with the passionate, keeps the third part, the animal, or appetitive, part of the soul, under control.
Socrates gives several arguments against his opponent, Thrasymachus, for why it is better to be just than it is to be unjust. I shall discuss these next.
I shall soon discuss this. Check back!