Generally, morality is a system of rules that modifies our behaviour in social situations. It's about the doing of good instead of harm, and it sets some standard of virtuous conduct. But if you look at the way in which moral values actually work in our everyday lives, you'll see that this is not the case.
Personal intuitions are important, of course.
But morality generally comes into play when people interact with each other. As such, morality is about deciding on best courses of action in all situations. If we did not have an values in common, it would be exceedingly difficult to agree on any one course of action. But since there is often disagreement as to what is the right thing to do in any situation, we can see that in fact, various values are shared to a greater or lesser extent. On some values there will be nearly unanimous agreement.