dispute
A serious disagreement where people argue about who is right.
A dispute is a disagreement or argument between people who see things differently. When two students dispute who won a race, they're each convinced they crossed the finish line first. When neighbors dispute where their property line ends, they disagree about who owns a strip of land between their yards.
Disputes can be small or large. You might have a brief dispute with your sister about whose turn it is to choose the TV show, or countries might have disputes over borders that last for years. What makes something a dispute rather than just a difference of opinion is that both sides care enough to argue their case.
Sometimes disputes get resolved through compromise or by bringing in someone neutral, like a teacher or judge, to decide. Other times people agree to disagree and move on. The word works as both a noun (you're in a dispute about the rules) and a verb (you dispute the referee's call).
Notice that disputing isn't the same as fighting: it's more about the back-and-forth of different viewpoints. Two historians might dispute what caused a war, each presenting evidence for their interpretation. That's different from just yelling at each other. The word suggests there's something real at stake and that both sides genuinely believe they're right.