Many people find a relativist account of ethics to be plausible. There are different versions of relativism, of course, and most people probably don't have a specific flavor of relativism in mind. What they are resistant to is a universal or objective account of morality.
In fact, many people take the debate to be settled. It's not infrequently that I'll hear someone declare, "Well, morality is purely subjective anyway, so..." They don't even assert this as their opinion. They're assuming that there's no fundamental moral truths, and seem to think it's the only reasonable thing to assume.
One motivation for this belief is a feeling that universalist accounts of ethics are arrogant and intolerant. It is the case that many people have done an awful lot of arrogant and intolerant things because they believed other communities had objectively wrong beliefs. They were bolstered in committing these injustices by a sense that they were objectively right. But the proper response to this is not to deny that actions can be objectively right or wrong. In fact, this ends up leaving us with no ground to stand on when we criticize the intolerant. The best response is to assert that it would have been objectively better if everyone had been less arrogant and more tolerant.
This relates closely to one of the points I made in my last post. Moral relativism is also often used as a defense mechanism for those facing moral criticism. If morality is relative, they will have no need to re-evaluate their views, and can dismiss the criticisms they are faced with. As I argued in that post, however, this defense is often directed as a moral criticism of their interlocutor, and so is self-defeating.
But even if some of the motivations for a relativist account of morality are dubious and self-defeating, this does not mean the account is incorrect. There are still some doubts about what kind of status moral claims have. As I've stated before, I believe the subject matter of morality is reasons, in particular normative reasons for action. Part of the argument for the objectivity of morality is an argument that people already think this way, that they deliberate by considering reasons, and that their beliefs about reasons make inherently universal claims.
Even if we accept all that, however, it might not convince us that claims about reasons are true. It will, at most, convince us that we cannot function without acting as though there are reasons. And still the worry persists, because reasons are a funny kind of thing. They are not like water, or the moon, or electrons. They're not even like the more interesting parts of the world, like shrubs, lizards, or human beings. In fact, we could likely give a full inventory of the entire universe and never use the word reason.
Daniel Dennett likes to say that this is because we're looking at the wrong level of investigation when we look for reasons in the same place we're looking for apples and carrots and centipedes. He's clearly on to something here, but talk of "levels" is a bit obscure. I worry that it merely sounds like it lifts more conceptual weight than it actually can.
A better way to think of morality, I think, is by analogy with something else that we won't find in our inventory of the universe. That is, time.
Like morality, many people have erroneously claimed that time is an illusion. Don't ask me for my theory of time, I don't have one. But it seems to me whatever you can say about time, it's not just an illusion. Events happen in time. Some things happened in the past, some more will happen in the future. These statements make sense, and are true. It seems to me this is as indubitable as any other claims we can make.
Yet people still question the existence of time. There exists a bias, I believe, of relatively recent origin, against truths that aren't the kind of truths you can express by banging on an object and asserting it exists. This bias is never argued for, just assumed. But of course we can say true things about time.
This is not to say that the metaphysics of time are obvious or simple. There is some interesting philosophical work done in this area and there's probably much of it that is too complicated for me to understand. But the thing we're not going to conclude after all this philosophy is that there's no such thing as time. It just, perhaps, needs a slightly more sophisticated account than that of tables and chairs.
Once we reflect on the nature of time, we can see that the mere fact that we don't find something among the planets, pebbles, or protons of the universe shouldn't necessarily cause us to doubt its veracity. Like time, morality is a concept that is, in part, about how we relate to the rest of the world. Its a feature of our experience of the world, which is why we shouldn't be surprised that its foundations aren't discoverable through science. But this doesn't give us any reason to doubt that it might be true.
Why do I focus on this issue? The tendencies towards relativistic, or even nihilistic, moral views in our culture lead to a troubling apathy. Most people don't appear to investigate their own moral beliefs to any depth. And many end up feeling so comfortable that most people share their moral views that they aren't even interested in whether or not their beliefs are true.
As a philosopher, I find this topic intrinsically interesting. But as an activist, I find this general apathy to be an indulgence to which I'm not privileged. I wouldn't be an activist if I didn't think it actually mattered what people do. If more people realized that there were truths about right and wrong, perhaps they'd be more interested to figure out the right thing to do.