Macaques and chimpanzees have a sense of social order and rules of expected behavior, mostly to do with the hierarchical natures of their societies, in which each member knows its own place. Young rhesus monkeys learn quickly how to behave, and occasionally get a finger or toe bitten off as punishment. Other primates also have a sense of reciprocity and fairness. They remember who did them favors and who did them wrong. Chimps are more likely to share food with those who have groomed them. Capuchin monkeys show their displeasure if given a smaller reward than a partner receives for performing the same task, like a piece of cucumber instead of a grape.
These four kinds of behavior — empathy, the ability to learn and follow social rules, reciprocity and peacemaking — are the basis of sociality.
Dr. de Waal sees human morality as having grown out of primate sociality, but with two extra levels of sophistication. People enforce their society’s moral codes much more rigorously with rewards, punishments and reputation building. They also apply a degree of judgment and reason, for which there are no parallels in animals.
From this NYTimes article by Nicholas Wade. The first part of the article lays out the evidence used in arguing about how morality might have evolved. But the second part is more interesting: it's here that philosophers get to air their objections to the evolutionary view of morality!