Moral realism and moral relativism give genuinely opposed answers to a basic question underneath ordinary ethical disagreement: is there such a thing as being simply, objectively wrong, or is "wrong" always relative to somebody's standards? Moral realism holds that at least some moral claims are objectively true, true independently of what any particular culture or individual happens to believe, in roughly the way a claim about the boiling point of water is true regardless of belief. Moral relativism holds instead that a moral claim's truth depends on the standards of the culture or individual actually making it, so the very same act can be genuinely wrong by one culture's standards and genuinely permissible by another's, without either culture simply being mistaken.
The two views disagree about what a moral disagreement between cultures actually is
When two cultures disagree about a moral question, a moral realist treats that disagreement the way they'd treat a factual disagreement about the natural world: at most one side can actually be right, and further evidence or argument could, in principle, settle who that is. A moral relativist treats the very same disagreement quite differently, as each culture correctly applying its own genuinely valid standards rather than as a dispute with one true, objectively correct answer waiting to be found, meaning the disagreement itself doesn't call for a determinate resolution the way a factual dispute would.
Each view faces a genuinely serious objection the other one is built to avoid
Moral realism faces the challenge of explaining what moral facts actually are and how anyone could come to know them, given that they don't seem to be discoverable through ordinary scientific observation the way facts about the physical world are. Moral relativism faces the opposite challenge: it seems to rule out ever genuinely criticising another culture's moral practices as actually wrong, since by relativism's own logic that culture's practices are correct relative to its own standards, a consequence many people find seriously counterintuitive when applied to practices they consider genuinely harmful. Each theory was built partly in response to problems the other one runs into, which is exactly why the debate between them has stayed a live, central one in ethics rather than settling either way.
What we're still unsure about
That moral realism and moral relativism give genuinely different, well-defined answers to what makes a moral claim true is well established, extensively debated territory within metaethics. What remains a genuinely live, unresolved philosophical question is which view, if either, is actually correct, since philosophers on both sides have developed serious, carefully argued positions and neither view has managed to fully answer the strongest objections raised against it, and this dispute continues to sit near the very centre of ongoing metaethical debate rather than being settled in either direction.
This sits inside Moral Relativism vs. Moral Realism, one of seven topics in Ethics, one of five domains in Philosophy, one of seventeen subjects the app can quiz you on.