For a realist, moral facts are as certain as mathematical facts. ("Moral Realism," The Internet Encyclopedia of Philosophy)
If "Stealing is wrong" is a true statement, then stealing is wrong for all people at all times. It is not wrong for some people or some groups and OK for other people or other groups. This is in contrast to moral relativism, which claims that moral statements may be valid for some groups but not for others.
The truth or falsity of moral judgments, or their justification, is not absolute or universal, but is relative to the traditions, convictions, or practices of a group of persons.
We can all agree that there are, in fact, many different and incompatible moral/ethical systems in the world ("descriptive moral relativism"). But most of us would like to think that there are some moral truths that are universally valid. "It is wrong to torture innocent children," for example. If we discovered a culture that tortures children for amusement, the relativist would find herself in a tight spot: if morals are relative to the culture, and this culture accepts torture of children, then on what basis can the relativist condemn this torture?
Moral realists insist that some things, like torturing children, are really, really wrong, independent of the culture or the individual's beliefs.
Some realists think that moral properties can be described in terms of some objective features of the natural world. This is called ethical naturalism. For instance, utilitarianism says that morality seeks the "greatest good for the greatest number of people." Assuming that the "greatest good" can somehow be quantified, this makes moral decisions reducible to calculations involving objective properties of the world.
For other realists, moral properties really exist but are irreducible properties. These are the non-naturalists. For instance, ethical intuitionists hold that humans have an intuitive grasp of right and wrong, and this intuition is not reducible to any natural features of the world (like pain and pleasure, for example).
Some other realist theories:
- Certain theistic theories, such as natural law theory.
- Deontological ethics: Right and wrong inhere in actions themselves, rather than in their consequences. For example, if killing is wrong, then killing Hitler is wrong, even if it would save millions of lives.
- Virtue ethics: Doing right consists of striving for excellence in all of one's roles: as (possibly) a parent, as a worker, as a citizen.