Moral relativism is the view that moral values and moral duties do not exist in reality, but only exist as opinions in people’s minds. When you ask a moral relativist where the belief that stealing is wrong comes from, he may tell you that it is his opinion, or that it is the opinion of most people in his society. But he cannot tell you that stealing is wrong independent of what people think, because morality (on moral relativism) is just personal preference.

So what’s wrong with it?

I found this list of the seven flaws of moral relativism at the Salvo magazine web site.

Moral relativists can’t accuse others of wrongdoing.
Relativists can’t complain about the problem of evil.
Relativists can’t place blame or accept praise.
Relativists can’t make charges of unfairness or injustice.
Relativists can’t improve their morality.
Relativists can’t hold meaningful moral discussions.
Relativists can’t promote the obligation of tolerance.
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I sometimes get flack from atheists who complain that I don’t let them make any moral statements without asking them first to ground morality on their worldview. And that’s because on atheism morality IS NOT rationally grounded, so they can’t answer. In an accidental universe, you can only describe people’s personal preferences or social customs, that vary by time and place. The answer is always going to be “it depends”. It depends on the person who is speaking because it’s a subjective claim, not an objective claim. There is no objective way we ought to behave.
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Without a designer of the universe, the question of how we ought to act is decided by people in different times and different places. It’s arbitrary and variable, and therefore it doesn’t do the job of prescribing behavior authoritatively. It’s very important not to get involved in any serious endeavor with another person or persons if they don’t have a sense of right and wrong being absolute and fixed. A belief in objective moral values is a necessary pre-requisite for integrity.