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Error theory basically states that all moral statements (e.g., "capital punishment is wrong") are false, because they all contain a false assumption. [not exactly: they rely for their truth on properties that don't exist]
The most widely known form of error theory comes from J.L. Mackie. Mackie argued that moral statements claim that the object of evaluation contains some sort of objective, intrinsic prescriptivity. For example, the statement, "Capital punishment is wrong," means, "Capital punishment contains an intrinsic property of ought-not-to-be-doneness." However, no intrinsic property of ought-not-to-be-doneness exists, so an assertion that capital punishment has this property must necessarily be false. [sort of; objectivity is defined -- by moral realists, not Mackie -- as a matter of intrinsic ought/ought not to be doneness; therefore, it is the property an anti-realist must reject/refute]
Similarly, "Everybody has an obligation to tell the truth," implies that truth-telling has an intrinsic property of ought-to-be-doneness. Intrinsic ought-to-be-doneness does not exist. Therefore, any assertion that truth has this property must be false.
Desire utilitarianism agrees with error theory that intrinsic ought-to-be-doneness and intrinsic ought-not-to-be-doneness do not exist [then it is a form of anti-realism by definition]. However, it denies that these elements are a part of the meaning of moral terms [meaning only that it is not a form of error theory]. "Capital punishment is wrong," does not mean "Capital punishment contains an intrinsic property of ought-not-to-be-doneness." Therefore, even though "Capital punishment contains an intrinsic property of ought-not-to-be-doneness" is false, "Capital punishment is wrong" is not necessarily false.
Arguments Against Objective Intrinsic Prescriptivity
J.L. Mackie gave two arguments against the idea that nature contained elements of objective, intrinsic prescriptivity; the Argument from Relativity and the Argument from Queerness
Argument from Relativity
The argument from relativity looks to the fact that different people in different cultures, and different people in the same culuture over time, have different opinions about right and wrong. One culture holds that women have the right to vote, own property, and engage in any action that a man may perform, while another denies women an education, a driver's license, or even the permission to be outside the home without a male escort.
It would seem that if objective, intrinsic prescriptivity exists, we should expect people from different cultures to see the same thing. This is true in the same sense that if trees exist, we would expect people from different cultures to see the same tree. However, the argument from relativity tells us that we do not have evidence of people seeing the same thing. This implies that there is no objective, intrinsic prescriptivity for them to see.
The argument from relativity is a notoriously poor argument. The fact that people disagree is no evidence that there is no right answer. If it were, then there would be no right answer about a great many things. [the argument is more complex than this; you are presenting a straw man of Mackie that is more akin to the anthropological arguments made by cultural relativists; Mackie's argument here, though, is epistemological]
People disagree about the age of the Earth. Some say that it is less than 10,000 years old. Others say that it is older than 4.5 billion years. If the Argument from Relativity were valid, then there would be no objective facts anywhere in the universe. Even the claim that there are no objective facts in the universe would be 'relative' in the sense that people disagree. [again, this is a strawman; mere disagreement is not the problem, but rather intractability and pervasiveness of disagreement even in light of the available facts]
One could respond that the Argument from Relativity does not hinge on the premise that people disagree on moral matters. Rather, it hinges on the premise that there is no objective way to resolve these differences.
This response fails on two levels. First, it begs the question. By assuming that there is no objective way to resolve differences in moral opinion it assumes that moral claims are not relative, which is precisely what the argument is supposed to prove. [it's an inductive argument, not a deductive one; you've missed the point entirely] Second, there is also no consensus over how to answer the qeustion of the age of the Earth. Some people argue that an appeal to scripture provides legitimate - in some cases, foolproof - evidence of the age of the Earth. Others claim it does not offer legitimate evidence. Either way, the question of 'how to resolve the dispute' is as much up in the air as the dispute itself.
If these matters yield the conclusion that there is no objective fact of the matter regarding the truth of moral claims, then they yield no objective fact of the matter regarding the age of the Earth either.
Yet, the question, "What is the age of the Earth?" does have an objective right answer. This shows that the argument from relativity is no argument against the objectivity of the subject in question.
Argument from Queerness
The Argument from Queerness says that queer (or bizaar) entities do not exist, and intrinsically prescriptive properties are queer. Take a proposed action. How is it even possible that the action could hold a property of intrinsic prescriptivity? What does it look like? How are we aware of its existence? How is it that this property compells either 'ought to be doneness' or 'ought not to be doneness?"
Taken literally, the 'argument from queerness' has some serious problems. Not the least of these problems is the fact that the universe seems to contain a large number of 'queer' entities, from black holes, to pulsars, to dark matter and dark energy, to nutrinos. The fact that these are very strange entities does not argue against their existence. [you're missing the nature of the strangeness; it's not just oddness, it's metaphysical incompatibility]
However, the question really is not one of 'queerness' as much as it is a question of 'usefulness'. We postulate black holes because we have theories that explain a great deal about nature, and those theories suggest that black holes do exist and that they have particular properties. When scientists turn their telescope into space these entities explain much of what they see and measure. The queerness of these entities takes a back seat to their usefulness. The queerness of intrinsic prescriptivity might take a back seat to its usefulness as well. [moral properties aren't useful in explanations in light of more parsimonious theories such as Mackie's; also, Mackie agrees that morality as an institution is useful, see the second half of his book]
Only 'objective intrinsic prescriptivity' is not useful. We can explain all elements of choice and the valuing of different preferences by talking only about desires, states of affairs, and the relationships between them. We do not need 'intrinsic prescriptivity'. Desires provide the motivational force for an action, not 'intrinsic prescriptivity'. [exactly Mackie's point]
There is nothing in the real world that gives us reason to believe that intrinsic prescriptivity exists. [again: that's exactly Mackie's point, which you have obviously missed]
Error Theory and Intrinsic Prescriptivity
J.L. Mackie's version of error theory says that all moral claims are false because (1) intrinsic prescriptivity is built into the meaning of our moral terms, and (2) all claims about intrinsic prescriptivity existing in any act or state of affairs is false.
One way to counter Mackie would be to argue that intrinsic prescriptivity does exist. However, moral realism can also be true, even if intrinsic prescriptivity does not exist, if intrinsic prescriptivity is not a part of the meaning of our moral terms. [no, it can't; moral realism requires the intrinsic prescriptivity; other forms of anti-realism don't, though]
The desire utilitarian alternative to Mackie's skepticism says that 'reasons for action that exist' are built into the meaning of all value terms. To call a state of affairs (for example) 'good' in the generic sense is to say that reasons for action exist for bringing about such a state of affiars. To call a state of affairs 'bad' is to say that reasons for action exist for avoiding such a state.
Intrinsic prescriptivity is one type of 'reason for action'. However, as Mackie argues, it is not a 'reason for action' that exists. As such, intrinsic prescriptivity cannot provide a real-world reason to engage in any real-world action.
Desires are the only reasons for action that exist. Desires take the form of propositional attitude - an attitude towards a proposition. By the very nature of a desire, it gives an agent a 'reason for action' to make or keep true the proposition that is the object of that desire. A person who has a desire to have sex with Jenny has a 'reason for action' for making or keeping the proposition 'I am having sex with Jenny' true.
Desires are reasons for action that exist. [yup, see part two of Mackie's book]
So, moral realism remains possible, even in the absence of intrinsic prescriptivity, [no, it doesn't; but first order ethics remains a possibility - which Mackie doesn't deny] of moral terms, like all value terms, refer to real reasons for action, and desires are real reasons for action.