Moral relativism is the philosophical position that morality is relative and that people should try to be good, but only by following their own consciences. Moral relativism can be contrasted with moral objectivism, the common position of many philosophers and religions that there is an objective morality, sometimes set down by God, an objective right and wrong. These two positions have tangled for thousands of years and are a contributing cause to many wars. However, it is arguable that wars and conflicts between conflicting notions of objective morality are more common than wars between objective and subjective moralists.
One phrase that partially sums up the philosophy of moral relativism is "live and let live." Sometimes the phrase "moral relativism" is used as a pejorative by moral objectivists and theists. This is often accompanied with the assertion that this relativism implies the complete absence of morality, but moral relativists do commonly believe in a moral code, just not that it is universally applicable. Among theists, moral relativism has a bad reputation, mostly because most religions teach moral objectivism. A prominent exception would be Buddhism.
Moral relativism has been around for a long time, with early writings by the Greek historian Herodotus (c. 484 – 420 BC) pointing out that each society has its own moral code and they all regard their own as the best. It should be noted that partial relativism is possible -- someone might believe in a core of objective moral truth, for instance, "killing is wrong," but believe that more nuanced issues, such as how much of one's income to give to charity, are more subjective. Most people, even self-labeled moral objectivists, usually have some area of moral reasoning on which they are not completely certain, and so concede some degree of moral relativism. Others would argue that this is not relinquishing moral objectivism, only admitting imperfect knowledge on what objective morality is.
One of the most famous and well-known moral relativist philosophers of the 20th century was Jean-Paul Sartre, who pioneered the philosophy of existentialism, which essentially asserts that mankind is alone in the universe and we have no morality to turn to except that which we create for ourselves. However, not all moral relativists agree with Sartre. Many moral relativists are simply motivated by the avoidance of ethnocentrism -- avoiding assuming that one's own culture is superior to others. They argue that this is essential for world peace, pointing to numerous historical examples when cultures served atrocities on others due to perceived moral inferiority.