This is a subject that is endlessly complicated, with much nuance and grey area. Morality and in particular moral authority. It is incredibly important to everyone with their everyday interactions and behaviors and yet hardly ever given conscious thought. There are many where we can see these values and beliefs are challenged and play out. From work, to play, to comment sections, political leanings and religious beliefs.
So let us get the easy one out the way. Theists have this one easy, relatively speaking. Their moral authority is their given to them from their deity or deities. From that point it becomes interpretation. Now that is neither simple or easy there and those issues are worth considering and weighing with due caution and diligence but at least you have a starting point there. So whilst this can be addressed in the future it may be a little more fun to muck around in even murkier waters.
There have been many great thinkers that have spent time in the waters surrounding what it is to be good including renowned philosophers like Immanuel Kant. These great people have written many thousands of words on the subject and many more will continue to do so. That is not what this is particularly about. I have no great revelation for you simply the question, “what conviction do you have to tell me what is right or wrong?” This is a question with no simple solution, no generic answer.
At certain stages in history we have decided how we, as human beings can act. One of the most important being the Nuremberg Trials. It was these places that we drew a very important line in the sand on what is acceptable as people. From there we have other authorities, the law, although one can argue that is of dubious morality at times, sometimes what is lawful is not morally acceptable (see the above mentioned Nazi’s). There are human rights codes, HR policies, parental teachings, codes of conducts and a number of secular ideological positions including veganism, libertarians, Marxists just to name a few of the many.
So I ask again, what conviction do you have to tell me what is right or wrong? and what makes you think I should accept it?