In Sentimental Rules (2004), Shaun Nichols does a good job of explicating the central Humean argument against moral objectivism.
As he summarises it, the argument broadly involves two steps:
1. Rational creatures who lack certain emotions would not make the moral judgments that we do.
2. There is no externally principled basis for maintaining that all rational creatures should have the emotions on which our moral judgments depend. (page 185)
We can imagine intelligent, rational, well-informed Martians who do not believe (for example) that it is wrong to torture puppies. They fail to condemn the practice of puppy-torture because they don't share the human emotional repertoire that leads us to revile it. At the same time, there is no independent basis for saying that our emotional repertoire is the "right" one and the Martian one is "wrong". Thus, our condemnation is not justified except relative to creatures with an emotional repertoire like ours. pp. 185-86) It cannot truly be said that the Martians are making an intellectual mistake, however much we may be appalled by them.
Nichols supports the first premise by referring to the range of empirical work that shows a link between affect and normative judgment (and, in his view, the cultural viability of moral norms). This makes it plausible that rational Martians with a different emotional repertoire would not have the same moral norms. The empirical evidence also tends to show that psychopaths, who do not have the normal emotional repertoire, make different moral judgments: for example, they do not distinguish between authority-contingent conventional norms and universal moral norms, and they give different explanations as to why harmful acts are wrong (for them, it as a matter of violating convention rather than of the welfare of the person harmed). (p. 186) As a further point, norms that are backed by emotions of disgust are more likely to be treated as akin to moral norms, which are backed by emotions of reactive distress and concern, than are norms that have no strong emotional backing, such as norms of etiquette on emotionally neutral matters such as table settings. (pp. 186-87)
The second premise can be filled out by saying that there is no principled, external reason why all rational creatures should have emotional responses such as the human responses of reactive distress and concern. It is not even clear how anyone could argue for such a claim (it is no use arguing in a circular fashion that Martians are morally obliged to have such responses, as determined by our moral norms and intuitions; this assumes what we set out to prove, that our moral judgments are objectively supported). It won't do to say that creatures without our emotional repertoire are in some way defective from an evolutionary viewpoint, since it is not necessarily the case that having such a repertoire would have been to their ancestors' evolutionary advantage. Even in the case of human beings, it is not clear that there is only one kind of emotional functioning that was evolutionarily adaptive. (pp. 187-88) It might be added that even if there were, this would not seem to settle the matter: why is it now objectively better to have an emotional repertoire that happened to confer a reproductive advantage in the environment of evolutionary adaptedness? Some further explanation seems needed.
Despite the long history of debate around this and similar arguments, I can see no way to avoid Nichols' conclusion - indeed, he has formulated a position very like the one that I find intuitively compelling. Nor can I see any way to avoid accepting the claim that this leads irresistibly to a conclusion that contradicts most commonsense and philosophical views of the nature of morality ("most", not necessarily "all").
None of this detracts from another set of facts - that most human beings do have similar emotional repertoires; that we readily end up embracing similar moral norms on a wide range of issues; and that we have reason to be glad of this, assuming everything else about our nature and situation is held fairly constant, and that we have certain widespread values such as preferring to live in a civilised society rather than a Hobbesian state of nature or something approaching it. There is a pragmatic viewpoint from which we can (almost) all consider our core moral norms justified, in that we want them to be retained even after we know all the facts, though we might question how far this extends beyond the core of morality to areas over which there is deep disagreement.
That last caveat raises important questions about how far a liberal society should enforce non-core moral norms that are often disputed, such as norms relating to sexual morality or reproductive practices. In one sense, these are no more non-objective than the core norms prohibiting cruelty and violence: i.e., none of them are objective in a hardline sense. However, those norms that plug strongly into the human emotional repertoire, and which forbid acts that cause pain or harm, are surely the most important to the survival of human societies and the least likely to cause resentment, opposition, and the infliction of suffering when they are imposed on the unwilling. To that extent, not all of our moral norms are equal. Legislators, please take note.