Moral relativism is not the same as moral nihilism. Moral nihilism is the view that there are no moral facts, whereas meta-ethical moral relativism, as I understand it, is the view that different cultures or societies have fundamentally different conceptions of the good and it is not possible to adjudicate which ones are more correct or better by any independent means. As a descriptive ethical philosophy, it is the mere observation that people in different cultures do disagree about morality. As a normative ethical philosophy, it is the view that moral principles exist only in particular social and cultural contexts and ought to be treated as binding within those contexts.
While moral relativism is a frequently criticized position (one might even say it is the ultimate boogeyman of many moral realists), it is difficult to find actual examples of people who defend moral relativism as a meta-ethical philosophy. Many people are relativists in the sense that they recognize (quite rightly) that different cultures have different ideas about how people ought to behave. But these sorts of relativists usually believe moral relativism is true only as a descriptive factual account and generally do not adhere to it as a meta-ethical philosophy.
The latter sorts of relativists, i.e. descriptive relativists, are generally relativists for a pragmatic reason: often, they are trying to understand the normative ethical worldviews of other cultures, and to do that they attempt to understand the values of other people in other cultures from their perspective. Of course, if they are savvy descriptive relativists, they recognize that there is also significant variation in values not just between different cultures, but also within them, from social group to social group, and at the smallest scale from individual to individual. But this issue, which cannot be understood without an analysis of power dynamics, conflict between groups, the creation and propagation of ideology, and efforts to achieve moral hegemony, is a discussion for a later time.
Normative moral relativism is problematic for the same reason as any other normative ethical theory which makes a universal claim of how people ought to behave: there are no objective or universal grounds for treating the moral norms of a particular group of people as binding on oneself when one mingles with that group’s members or goes to a place they inhabit or frequent.
In fact, in many situations treating the moral norms of a particular group as binding when one is in their presence or in their territory would require a person to compromise his or her core values, such as when a person of an egalitarian and humanistic bent spends time with members of a group the majority of whom condone cruelty towards other people. In other words, it must be acknowledged that the values a person believes in can and often will conflict with the dominant values of at least some other group of people, and at that point it is necessary to either renounce one’s own values in favor of embracing that other group’s values, as the normative moral relativist would have one do, or reject moral relativism and instead seek to live by one’s chosen values.
There is one common form of moral relativism that many people accept as valid, but few recognize as a form of moral relativism: namely, the view that we ought to judge people in the past by the standards of their time rather than our own standards or any alleged universal standards. The natural initial response to this is “whose standards?” For example, do we judge whether slavery was wrong by the standards of slaveholders and defenders of slavery or the standards of slaves and abolitionists? But even if we assumed that the standards of past societies were monolithic (even though they of course never were and in fact, like today, were always changing), there is still no justification for not judging past societies by contemporary moral standards, at least as a normative matter, if those standards are objective and universal. As a descriptive matter, it makes sense to utilize relativism the same way anthropologists (ideally) do when conducting ethnographic research, that is, in the way described above.
Another problem with judging people in the past by the standards of their time, at least in its application, is that people generally show incredible leniency when judging people from their own culture or society’s past who they admire, but are generally very critical of people from other culture or society’s pasts. Just as a way of illustrating this, I would ask rhetorically, how many people calling for Confederate leaders like Jefferson Davis and Robert E. Lee to be judged by the standards of their time would advocate the same treatment of Muhammad and his early followers? (To be clear, I advocate that all people in the past be judged by any applicable universal standards, though as I do not believe universal standards exist, I judge them subjectively.)