A section on morality on a Christian apologetics website might be surprising, but it should not be. When it comes to right and wrong, we need to offer wise counsel. After all, people have objections about God, people's actions, political issues, etc. Most always, the issue deals with what is right and wrong.
Without a standard of morality, there is no way to judge what is good or bad. Atheists, for example, might decry what is the behavior of God in the Old Testament when he orders the destruction of people groups. But, by what standard does any atheist have to judge what is morally correct? At best, an atheist would only have the ability to express an opinion since he cannot offer any objective standard of morality.
Religious people can appeal to a higher power from which they can ascertain what is good and bad. Of course, Christianity doesn’t recognize the existence of any gods other than the one taught in Scripture. For Christians, God's revelation found in the Bible is the standard by which they make such judgments. So, to answer the question about what gives us the right to judge what is moral, the answer is God. We don't judge what is right and wrong. God does, and he has communicated his standard of righteousness. This communication is found in the Bible. Take the 10 Commandments in Exodus 20. We see a codification of moral standards. We are told not to lie, not to commit adultery, not to covet, etc. These are standards given to us by God, and though there are other cultures that don't believe in the biblical God, they might have similar moral codes. But, for the Christian, the Bible is the supreme authority that judges what is moral.
We can only judge what is moral if we have a standard given to us by God, not some standard that is based on emotion, opinion, or the changing morals of society. Even though atheists, agnostics, Muslims, and non-Christians might not approve of standards found in the Scriptures, we Christians believe that the Bible is the revealed and inspired Word of God and that within its pages are the moral standards by which we are to model our behavior. Therefore, the right we have to judge what is moral comes from God as is revealed in his Word.
Finally, we must be extremely careful as Christians to apply moral judgments with equity, non-partiality, love, and truth. Moral judgments without compassion and mercy become immoral in themselves. This is why we are to keep our eyes on Jesus who is the judge and the standard of righteousness. We must keep our eyes on his words and his deeds and emulate him. Jesus forgave, was patient, taught, and healed. His words were not simple moral statements. They were manifested by his life and loving character.