When people argue, they disagree with each other, sometimes loudly. You don’t always have to yell when you argue, though, sometimes you’re just trying to prove a point. If your mom tells you to be home for dinner, don’t argue!
When you think of the word argue, you might think of angry people screaming at one another, but that’s only one meaning of this verb. The other is "to present reasons and evidence about a stance or opinion." For example, if you give a speech in support of world peace, you might argue that nations accomplish more when they work together, giving examples to support this idea. When you present reasons why you believe in something, you argue for it.