Arguments against moral relativism:
According to moral relativism, there is no objective standard of goodness that can be made. Thus, rights and wrongs are therefore only relative to certain cultures. That means, that things such as slavery, rape, or torture is not really "wrong", but it is only wrong for our culture, and it is right for another culture just because that culture thinks it's right, so we have to disregard everything as to "why" we think it's wrong, even though that culture's viewpoint ignores the experience of the victims, and even though the people within that culture who committed the acts themselves would not like or find it "right" for them to be acted upon in such a way.
The change argument:
"Sometimes our view about the moral status of some practice changes...When a person's moral views change in this fashion, the do not merely drop one moral belief in favor of another. Typically, they also hold that their previous moral view was mistaken. They take themselves to have discovered something new about what is morally right. Likewise, then the prevalent moral belief in a society undergoes a significant change, as in the civil rights movement, we are inclined to see this as a change for the better. But the relativist cannot account for changes in our moral beliefs being changes for the better. This is because the relativist recognizes no independent standard of goodness against which the prevalent moral beliefs after a change can be judged to be better than the prevalent moral beliefs before a change."
source:
http://facweb.bcc.ctc.edu/wpayne/Moral%20Relativism.htm
"Not only does moral relativism entail that we cannot make legitimate moral comparisons of different cultures, it also entails that we cannot make legitimate moral comparisons of a single culture across time; we cannot judge whether a changing society is getting better or worse. Generally, though, we do think that we have made moral progress. Moral relativism, arguably, cannot make sense of this."
source:
http://www.moralrelativism.info/moralprogress.html
A case for an objective morality:
The Great reformer argument:
"A final argument against cultural relativism is that cannot account for the existence of great moral reformers. When we consider those people that have helped to bring about those changes that we take to constitute moral progress, e.g. the abolition of slavery, or granting the working classes and women the right to vote, we generally think these reformers are moral exemplars, excellent people.
According to cultural relativism, though, these great reformers were bad people. According to cultural relativism, moral goodness consists in acting in the ways prescribed by the values of one's own culture. Those who seek to change those values, then, are bad people. Martin Luther King, Emily Pankhurst, and Gandhi, all of whom opposed existing values and sought to improve them, must all be judged by cultural relativists to have acted immorally. Those who we tend to think of as heroes must, if cultural relativism is true, be condemned."
source:
http://www.moralrelativism.info/greatreformers.html
Objective Morality:
An objective morality is a standard of morals that transcends an exclusive subjective preference. It means that everyone must be considered of equal value. This is what many of us think of when we think of morality.