The difference here is that moral relativism says that there is no fixed morality whereas moral absolutism holds that there is. The latter holds that some things are absolutely morally good or bad while the former holds that things are only morally good or bad in context.
For example, let us look at the idea of chicken fighting as animal abuse. Here in the United States, this practice is seen as immoral because it is cruel to animals. To a moral absolutist, this would mean that chicken fighting is immoral whether done by an American in the US or by a Filipino in the Philippines (my uncle in the Philippines raises fighting cocks). To a relativist, the American would be acting immorally because his culture says chicken fighting is immoral. However, the Filipino would not be acting immorally because his culture says that it is acceptable.
This is the difference -- relativism says that what's moral changes from place to place while absolutism says that something that is truly immoral is immoral no matter what a particular culture thinks.
Moral absolutism and moral relativism are essentially opposite approaches to thinking about ethics and morality. Moral absolutists hold that some things are absolutely right or wrong no matter the circumstances. Moral relativists, on the other hand, believe that under some circumstances, some things might be morally acceptable, but not in others.
What do we mean by "circumstances?" Relativists might suggest that morality is situational. Where a moral absolutist would hold that it is always wrong to lie, a relativist might argue that under some circumstances, lying is the moral thing to do, for instance if one was lying to protect another person. Stealing might be morally wrong under some circumstances, but stealing food to feed the hungry might be acceptable. Even war or killing might be acceptable if it is fought in self-defense, or in pursuit of some greater end.
Another facet to relativism, one which is less philosophical than these concerns, is related to different cultures. What is morally acceptable in some cultures might not be in others. We often see this when Western values clash with those of other peoples around the world. In this light, moral relativists tend to emphasize that citing one's own morality as absolute entails dismissing the cultures of other peoples. As the world has become increasingly globalized, and nations more ethnically and culturally diverse, this issue has become a major dilemma for many societies.