Before reading this article, check out What is Relativism.
The basic definition of moral relativism is that there are no moral truths.
Many people believe that moral relativism is a logical consequence of human reasoning. The basic idea is that all humans are morally equal so none of us have the authority to tell anyone else that this or that is morally true.
The problem is that logic, and the behavior of moral relativists, destroy their own argument very quickly. Let's demonstrate.
Person 1: "Artificial contraception is truly immoral."
Person 2: "That's not the truth; it's your opinion."
See what happened? Person 2 said Person 1 is wrong. Why; because Person 1 violated moral relativism. The only way Person 1 can violate moral relativism is if he is subject to it. The only way he can be subject to it is if it applies to all people and the only for that to happen is for moral relativism to be true.
In brief; "there are no moral truths" is self-defeating because it is a moral truth.
This is like saying "I do not exist". A logical question is "how did I write that sentence if I don't exist?" Another logical question is "if moral relativism is a moral truth, how can moral truths not exist?"
Moral relativism isn't logical; in fact it destroys itself quite conveniently if we care to reason it out.