I worry that we are lacking consistent terminology here on this thread. For one thing, moral objectivism and moral relativism are contraries, but not contradictories (that is, you can’t hold both, but you don’t have to hold one or the other). Perhaps some standardized definitions are in order.
Moral realism is the technical term for what most people call moral objectivism. To be a moral realist, you must assert all three of the following:
(1) Moral statements are the kind of thing that can be true or false.
(2) Some moral statements are true.
(3) True moral statements have their truth value in virtue of the metaphysical status of the actions they describe (that is, something about the metaphysics of the universe makes moral statements true).
Note that these theses are not entirely independent. If you deny (1), you must also deny (2) and (3) on pain of contradiction. If you deny (2), you must also deny (3). Only (3) may be denied without denying either of the remaining theses.
Moral anti-realism is the contradictory of moral realism. To be a moral anti-realist, you must deny one of the three theses of moral realism. Given that the three theses of moral anti-realism are not entirely independent (as noted above), all moral anti-realists deny (3).
Constructivists only deny (3). They hold that there are true moral statements, but that the attitudes of sentient beings are in one way or another essential to the truth of those statements. There are many varieties of constructivism, some of which are very close to moral realism.
The most common, however, are moral subjectivism (which holds that moral statements are assertions about personal attitudes and thus are true when the person asserting them accurately conveys his attitudes) and moral relativism (which holds that moral statements are assertions about group attitudes and thus are true when they actively reflect the attitudes of the group from which the statement is being made).
Another common constructivist view is called ideal observer theory, which holds that the best thing to do is what a perfectly rational and fully informed individual would tell us to do. Divine command theory, which says that we should do whatever God commands, is one version of this view (placing God in the place of the ideal observer). As divine command theory is a form of constructivism, it is also a form of moral anti-realism.
Error theorists deny both (2) and (3), holding that moral statements are failed attempts to refer to moral properties that do not exist. Thus moral statements are the kind of thing that can be true or false, but it turns out that none of them are true because the world lacks anything to make them true. Error theories are always relative to a particular moral discourse. So while one might think that ordinary moral discourse is erroneous, such an error theorist is not committed to the view that all possible moral discourses are erroneous.
Non-cognitivists deny all three theses of moral realism. On this view, moral statements are not assertions at all—despite their syntactic structure—but either commands or effusions of emotion. They do not need anything to make them true because they are not attempts to say anything that is true, informative, or descriptive. Instead, moral statements are solely in the business of getting people to do things with no additional semantic content whatsoever.
This is by no means a complete taxonomy of ethics, but it covers many of the most relevant positions. I have used the terms “moral realism” and “moral anti-realism” to avoid confusion with Ayn Rand’s so-called “Objectivism,” which is not what most people mean when discussing objective ethics. In fact, Rand rejected moral realism and called it not “moral objectivism,” but rather “moral intrinsicism.”