My recent post detailing two anti-realist options better than relativism has attracted much discussion. I can understand if you think my focus on setting up these distinctions only arises because when it comes to ethics,
But it's not just that. The metaethical theory you accept -- whether it be error theory, noncognitivism, relativism, or objective realism -- has enormous implications for what you can say is right and wrong. I'm thinking now that I haven't yet presented the argument against relativism that would appeal most to liberals. This is going to be similar to my argument against speaker-relativism.
Consider the situation of the moral reformer -- the person who thinks her culture is accepting the wrong set of moral norms. Maybe this person is in the society prescribed by Deuteronomy 22:21. She says, "People here think it's okay to stone women to death for having had premarital sex, but it's actually horribly wrong." If you go relativist and consider moral claims to be about what the speaker or agent's culture regards as moral, you get this translation of her utterance: "People around here think it's okay to stone women to death for having had premarital sex, but actually people think it's completely forbidden." This is a straight-out contradiction. If you take the moral facts to be, by definition, facts about what society permits or forbids, the moral reformer is screwed the moment she opens her mouth. In espousing a moral system that isn't that of her culture, she's doomed to be wrong. Please, please don't go relativist. It changes the rules of moral argument into something that can be utterly vicious.
Perhaps most people who identify as "relativists" don't want this outcome. As Angelica and Tony pointed out in comments, the term "relativism" broke its cage in the philosophy lab a long time ago and is running wild in the culture. Etymology probably doesn't have control over the term at this point, but it gets its meaning from the idea that moral demands are only true relative to a particular speaker or agent, or her cultural surroundings. And I'd say that in the broader culture, its connection to cultural norms as the truth-makers for moral claims remains intact. Insofar as this is the case, it's something that must be resisted if moral progress is even to be regarded as conceptually possible.
Error theory and noncognitivism, respectively, are two ways of capturing what often motivates relativism without giving a weird theory of moral terms that puts the reformer into this position. Now, error theory isn't good from the perspective of anyone who's rooting for moral improvement -- since all moral claims are false, it's hard to see how an error theorist can be consistent in morally evaluating others' behavior as right or wrong. Perhaps the error theorist can have something else that it's right to promote (happiness? desire-satisfaction?) which she doesn't identify as what morality is about. And error theory doesn't make the moral reformer necessarily immoral, since it holds that there isn't any real morality or immorality. While I think noncognitivism is implausible because of its inability to successfully translate some more complicated moral utterances, its more recent and sophisticated versions survive many tests that relativism does not. If people are interested and I'm sober sometime, I'll post on varieties of objective realism. There are ways of running realism that don't commit you to anything beyond the scientific world-view, and don't require you to mangle moral concepts nearly as much as relativism does.
The broader point here is that our views on morality, or something like it, are going to inform a whole lot of how we act, vote, talk about politics, and operate in the world. We care about morality a lot, and it matters that we understand what it is. Our views on the nature of moral concepts have big consequences for the theory of what's right and wrong. So we'd do well to clarify them and make sure they're what they ought to be.