Moral realism is, in essence, the concept that moral statements deal with a real thing. That is to say, a statement such as "murder is wrong" is a statement of fact in much the same way that "the sky is blue" is. As a proposition, it could be said "the statement "murder is wrong" is true." Generally speaking, moral realism is the default position in most societies today. The assumption is that when one makes a moral statement, they are making a statement of fact. Our legal system attempts to find the truth of morality, and we condemn others when their morals are incorrect. Others assert this is nonsense, and multiple arguments are employed to that end. While I do understand the reasons one would argue in favour of moral realism, and while I intend to present them fairly, I remain unconvinced by them.
The first argument for moral realism is a descriptive one: people already use a realistic framework. Consider a preference, say for food. Perhaps you do not like the taste of broccoli. You can imagine an alternate version of yourself, exactly the same except for a love of broccoli. Imagining this is probably not disturbing. Though you may not care for broccoli, you are not bothered by the possibility of that changing. Consider instead a moral preference. Likely, you do not look fondly on slavery. If you were to have been born some 300 years earlier, it is quite likely that you would see no problem with slavery Most people have a much stronger reaction to a version of them with different morals than a version with different preferences. It is then argued that this is evidence that morals are not merely preference. It is objectively wrong to support slavery. I take an alternate explanation: moral preferences are part of your identity. To change your morals is to change who you are, which explains why it is more upsetting to consider. One could develop a taste for broccoli and still be otherwise exactly the same. But for most people, slavery conflicts with a great deal of other deeply held ideas--you'd have to change who you are. Here is a prescriptive argument for moral realism: some sense of objective morality is necessary for society to function. The justice system cannot operate on a purely relativistic framework, because it must dole out punishment according to some set of standards. If everyone had wildly different ideas about morality or did not believe in morality at all, everything would fall apart. While this is true, it does not change my feeing. In much the same way that one might argue belief in god is a good idea (Pascal's wager), I still cannot induce myself to believe.
There are also a few positive arguments against moral reality. The most convincing, to me, is that morality is evidently immeasurable. A framework purporting a thing to be real, but being unable to measure that thing is useless. If it is the case that a moral belief can be incorrect, there is no means by which to determine whether it is. In other words, the truth value of a moral proposition cannot be known. From this point, there are a few different paths. Error theorists hold that moral propositions attempt to express truth, or, in other words, that a moral claim is a statement of fact. However, error theorists state that the truth value of those claims are always false. Therefore, anyone making a moral claim is in error to do so. Emotivists, instead, argue that moral claims do not attempt to express truth. Instead, they express emotion. Saying "murder is wrong" is the same as saying "boo! murder." It is neither true nor false. I align more with emotivism, as it seems to better describe the function of moral claims. It does appear to be the case that moral claims express emotion primarily.