The first clear statement of relativism comes with the Sophist Protagoras, as quoted by Plato, "The way things appear to me, in that way they exist for me; and the way things appears to you, in that way they exist for you" (Theaetetus 152a). Thus, however I see things, that is actually true -- for me. If you see things differently, then that is true -- for you. There is no separate or objective truth apart from how each individual happens to see things. Consequently, Protagoras says that there is no such thing as falsehood. Unfortunately, this would make Protagoras's own profession meaningless, since his business is to teach people how to persuade others of their own beliefs. It would be strange to tell others that what they believe is true but that they should accept what you say nevertheless. So Protagoras qualified his doctrine: while whatever anyone believes is true, things that some people believe may be better than what others believe.
Plato thought that such a qualification reveals the inconsistency of the whole doctrine. His basic argument against relativism is called the "Turning the Tables" (Peritropé, "turning around") argument, and it goes something like this: "If the way things appear to me, in that way they exist for me, and the way things appears to you, in that way they exist for you, then it appears to me that your whole doctrine is false." Since anything that appears to me is true, then it must be true that Protagoras is wrong . Relativism thus has the strange logical property of not being able to deny the truth of its own contradiction. Indeed, if Protagoras says that there is no falsehood, then he cannot say that the opposite, the contradiction, of his own doctrine is false. Protagoras wants to have it both ways -- that there is no falsehood but that the denial of what he says is false -- and that is typical of relativism. And if we say that relativism simply means that whatever I believe is nobody else's business, then there is no reason why I should tell anybody else what I believe, since it is then none of my business to influence their beliefs.
So then, why bother even stating relativism if it cannot be used to deny opposing views? Protagoras's own way out that his view must be "better" doesn't make any sense either: better than what? Better than opposing views? But there are no opposing views, by relativism's own principle. And even if we can identify opposing views -- taking contradiction and falsehood seriously -- what is "better" supposed to mean? Saying that one thing is "better" than another is always going to involve some claim about what is actually good, desirable, worthy, beneficial, etc. What is "better" is supposed to produce more of what is a good, desirable, worthy, beneficial, etc.; but no such claims make any sense unless it is...