The law of non-contradiction states, a thing can not both be and not be at the same time and in the same respect.  We can thank Aristotle for discovering it for us. The law of non-contradiction can be applied to a relativist's statement regarding one's understanding of either one's existence, or of existence itself.  In symbols, the law is written, A is not -A. 

Let's take a scenario in a freshmen Philosophy 101 class.  
Jason says, "All things are relative to me." But, Eric is not so moved by this appeal to relativism. Eric may say, "All things are not relative to Jason." What has happened now is that Eric has contradicted Jason. However, Jason replies that all things are relative to him, and that for Eric, things are relative to him, and that these are two, separate claims regarding two, separate people and therefore two, separate interpretations of reality. Not so fast. Eric is a sharp one. He says, "To the contrary. You have stated that it is a fact that all things are relative to you (which means what, exactly?--we shall get to this), and I am contradicting what you say. And because I am contradicting you, you must be assured that your relativism is falsified, because the law of non-contradiction is at work."
Eric is correct. This is true, because, on individualist relativism, each particular person is the creator of metaphysical states of affairs (truths, facts, etc.).  Or, each person is the single, particular creator of truth regarding what is real. On that score, it would seem impossible for another to come along and contradict him, because a single truth-claimant is in a vacuum.  But if the truth-claimant is in a vacuum, then he is alone by himself and therefore cannot say anything about "all things" for the "all things" to which he points are his perception of extra mental reality. He would only be able to say, "I am relative to me," which is absurd. This would be like going the solipsist route and saying all is imaginary to him, and that he is the only one who exists. But this is absurd, because then one thing is true--and not in a relative sense--that he exists. Or, he could deny his existence and say that all is imaginary, but this too, is absurd. One cannot deny one's own existence without simultaneously affirming it along with the denial.
"I do not exist." Right.
So, it is impossible for Jason to say "All things are relative to me" because this means that either he is a solipsist, or in a vacuum; and if he is in a vacuum, he is unable to communicate to others.  To whom is he communicating if he is in a vacuum?  "All things are relative to me," he says. And to whom is he speaking? Another person who truly exists? And if the other person is imaginary (to the mind of Jason), then Jason most certainly exists, and all things are not relative to him, for he is a part of everything, and he himself cannot be relative, because this is a self-referentially absurd statement e.g. "I do not exist."
Now, how is the law of non-contradiction at work? Eric contradicts Jason. We now have a contradiction that exists.  Jason says, "It is true that, 'All things are relative to me'" and Eric says, "It is not the case that 'All things are relative to Jason.'" Or, he could say, "It is false when Jason says, 'All things are relative me.'" Now Jason must respond by either repeating his axiom ad infinitum and beat his fist on the table, or he must give evidence for his belief. We have seen that he can give no evidence for his belief, because his only option is a vacuum or solipsism (if there is much difference between the two). Now, if Jason simply beats his fist on the table and repeats himself, he is no longer doing philosophy. He is a 'mutist.' As Frederick Wilhelmsen was quoted earlier, a civilization where the denial of reality is rampant, is one that is in decay.

 So we have the law of non-contradiction at work: Jason says A, and Eric says not-A. The law of non-contradiction transcends these two statements and rules over them, demanding an answer of the truth of the matter. So now Jason is in a real pickle: he cannot deny that the law of non-contradiction exists, because it is plain that he has been contradicted. What will he do? He can deny that the contradiction exists, but then he would be affirming the following:
"I deny that which Eric denies, which is that all things are relative to me." Now, to deny a proposition as true is to affirm it as untrue, and to affirm it as untrue is to contradict the proposition. Denying Eric's proposition requires contradicting it, which requires the law of non-contradiction to be at work. The law of non-contradiction cannot be denied without being used.  Therefore, the law of non-contradiction is a real law of being--and existence--showing us that things are truly real and not relative. Therefore what Jason says is absurd.
Perhaps Jason is a mutist and fears the absolute, because, as Wilhelmsen says, the absolute will mock his way of life: if the absolute is real, maybe there is a standard by which Jason will be judged--and he doesn't want that. Or, perhaps Jason not avoiding any moral issue, but is simply misguided for one reason or another (a high school teacher, perhaps). Another option is that Jason is just playing philosophical games for his own amusement's sake (and possibly for the sake of others).  He really does believe reality is absolute, but is just having some fun. Or, finally, perhaps Jason is proposing relativism in order to stimulate good philosophical discussion. That will be for Jason to decide which is true, which is the case, which is real.