I’ll try to put this as succinctly as possible while still doing it justice (though bear in mind that much more qualified people have written whole books on the subject).
There are really two types of relativism: absolute relativism and moral relativism. Absolute relativism says that “there is no objective truth.” This of course is self-refuting, because that is an absolute statement; the speaker has made an objective claim. It is patently false, just as the assertion, “this is a false statement.” In order to be true, the statement must be false! Therefore absolute relativism cannot stand on its own two feet.
Moral relativism is different. When one says “there are no moral absolutes,” that person has not made a moral claim, but a claim
about morals. Since it is not a moral statement itself, it cannot be self-refuting. I feel that the surest way to refute moral relativism is to draw out its implications, which its adherents often don’t even think about.
Now some truths are indeed relative. I’m right in saying that chocolate ice cream is great. You, however, might loathe the stuff, and instead say vanilla is great, which
I hate. Our views conflict, but we’re both right, because there is no objective value in the stuff. The problem is that some people treat morals as they do flavors of ice cream. You might not like to steal, but I do. Who’s to say who is right and who is wrong?
But if you take that same view further, then you end up with people that want to know what’s wrong with them torturing babies. It’s just what they happen to like! And so with such a view of morals there is a complete moral equivalence between Mother Theresa and Adolf Hitler.
This is what a true moral relativist must admit.
Briefly, I’d like to mention the “seven fatal flaws” of relativism from the book I mentioned earlier by Koukl and Beckwith:
1: Relativists can’t accuse others of wrongdoing
2: Relativists can’t complain about the problem of evil
3: Relativists can’t blame or accept praise
4: Relativists can’t make charges of unfairness or injustice
5: Relativists can’t improve their morality
6: Relativists can’t hold meaningful moral discussions
7: Relativists can’t promote the obligation of tolerance
Let me know if you want any of those fleshed out!
Hope this helped.