Non-cognitivism in ethics is the idea that moral claims are really just expressing the speaker's attitude towards something. "Abortion is wrong," for example, amounts to "Boo on abortion!" The attitude expressed is, roughly, "I approve/disapprove of this and you should, too." But the moral claim is itself the expression of the attitude - it is not the claim that one has that attitude. That is, a moral claim is non-propositional: it doesn't have any content that is capable of being true or false.
Cognitivist philosophers respond that we seem to think moral claims have propositional content. They support this by considering the way we use moral language. Consider the claim, "If killing animals is wrong, one shouldn't eat meat." It doesn't make any sense to translate this as, "If boo on killing animals!, one shouldn't eat meat."
I think the cognitivists are probably right: when we make moral claims, we think we are making a statement that is capable of being true or false. (Whether we are right to think so is another matter.) But it seems to me that once they have considered and dismissed non-cognitivism, cognitivist philosophers forget all about the non-cognitive aspect of morality.
In the last post I gave a sketch of morality as an evolved social system that limits the actions of individuals by means of social pressure. If this view is at all correct, it is easy to see why there is a large non-cognitive component to moral claims. A strong expression of disapproval of doing X, and the resultant peer pressure to refrain from doing X, lies at the root of the moral system. (Together with the positive version: expressing strong approval of an action.)
It's harder to understand why there might be a cognitive component to morality. It seems like evolution could have given us a purely emotional response that would serve to enforce the group norms.
Let's see if the linguistic analogy helps us here. Consider "It's wrong to say, 'I am going the store to.'" Here "wrong" is not used in the moral sense, but it plays a similar role. It certainly expresses disapproval. But it also implicity invokes the rules of grammar: "It's wrong to say, 'I am going the store to,' because in English the preposition comes before its object." Notice that in grammar, the rules arise (originally) as generalizations about actual usage. They are not imposed by some linguistic authority - though various institutions (dictionaries, textbooks) might take on that authority at a later time.
It seems to me that moral claims are also two-pronged: they express approval/disapproval, but they also implicitly invoke general rules that it is assumed are accepted, or at least known, by all. "Murder is wrong" thus has the cognitive content "Murder is a violation of the generally accepted rules of behavior."