"Isn't all morality relative?"
I often get this question from students, as there seems to be a fairly pervasive acceptance of moral relativism in much of our culture these days. However, the answer to this question is pretty clearly "no."
There are a variety of philosophical arguments against moral relativism. Some of them are reasons for accepting moral realism, which is the view that there are some objective moral truths. Other arguments against relativism point out some of the problematic implications it has, as well as the flaws in arguments that have been offered in favor of relativism.
First, consider that one powerful argument in favor of moral realism involves pointing out certain objective moral truths. For example, "Cruelty for its own sake is wrong," "Torturing people for fun is wrong (as is rape, genocide, and racism)," "Compassion is a virtue," and "Parents ought to care for their children." A bit of thought here, and one can produce quite a list. If you are really a moral relativist, then you have to reject all of the above claims. And this an undesirable position to occupy, both philosophically and personally.
Second, consider a flaw in one of the arguments given on behalf of moral relativism. Some argue that given the extent of disagreement about moral issues, it follows that there are no objective moral truths. But this is what a basic logic text refers to as a non-sequitur. The conclusion does not follow from the premise. To see why, consider a different argument of the very same logical form. There is extensive disagreement about the existence and nature of God, therefore it follows that there is no truth of the matter about God's existence and nature. But there is a truth of the matter. There either is a God or not, and if there is then that God is perhaps the Judeo-Christian God, or the Muslim God, or perhaps satisfies a conception of God we are as yet unaware of. But there is a fact of the matter, even if we don't know what it is, or fail to agree about it. Similarly for morality, or any other subject. Mere disagreement, however widespread, does not entail that there is no truth about that subject. It might be difficult to ascertain, and we may never reach full agreement, but that is different than the claim that there is no such truth to pursue.