Distinguish two things that constantly get merged. Descriptive relativism is the observation that moral beliefs differ across cultures — an empirical fact nobody disputes. Normative relativism is the philosophical claim that there is no framework-independent standard by which those beliefs can be assessed. The first does not entail the second. People also disagree about the shape of the earth, and we do not conclude that its shape is relative to culture.
Protagoras gave the classic formulation, holding that the human being is the measure of all things, and Plato pressed the objection that has followed relativism ever since. If Protagoras' doctrine is true, and most people believe it false, then by his own standard it is false for them — so the doctrine undermines itself. The pattern recurs in every version: the claim that all truth is framework-relative seems to be asserted as a truth that is not.
Relativists have replies. The most common is to make the claim self-consciously relative — true within the framework of the relativist — though critics note this drains it of interest, since it becomes an autobiographical report rather than a position. A more careful move restricts the scope: relativism about morality or aesthetics need not be relativism about everything, and a relativist about moral claims can hold ordinary factual claims absolutely without contradiction. That is the strongest version of the position, and it is not obviously self-refuting.
The tolerance problemWhy it doesn't do what people want it to.
Relativism is often reached for as a basis for tolerance, and this is where it turns on its adopter. If moral claims are true only relative to a framework, then tolerance is good is also only true relative to a framework — and cannot be recommended to a culture whose framework rejects it. Worse, the position removes the standing to criticise anything: a practice you find abhorrent is, by hypothesis, correct within its own framework, and your objection is merely a report of your framework's contrary view. Relativism is not a tolerant doctrine; it is a doctrine that makes tolerance unrecommendable along with everything else.
It also has trouble explaining moral progress. Most people believe that abolishing slavery was an improvement, not merely a change. On a strict relativist account there is no framework-independent standard against which to call it progress — only one set of local standards replacing another. Some relativists accept this; most find it a heavy cost.
The strongest caseWhat relativism gets right.
It would be unfair to leave it there, because relativism is responding to something real. Moral disagreement is deep and persistent in a way disagreement about chemistry is not, and no proposed method has resolved it. Our confidence in our own moral views is substantially explained by where and when we were born, which should induce some humility. And attempts to derive objective moral facts have not produced consensus either — the realist's position is not obviously in better shape than the relativist's. The honest summary is that relativism has a serious problem with self-refutation and tolerance, and that the alternative has serious problems of its own.