If kids can't settle their own argument, they may need someone to mediate their dispute, or conflict. You can often tell this is the case if you hear the loud cry, "Mom!"
The noun dispute means any conflict or disagreement. The countries were in the middle of a border dispute; both sides claimed the toxic waste dump belonged to the other. As a verb, dispute can mean to quarrel or argue, but it can also mean to take exception to something. If your boasting friend crows because you lost the trivia game to her by 150 points, you may want to dispute the facts. It was only 145 points.