When two people disagree, they have completely different opinions about something. You might disagree with your sister about whether chocolate or vanilla ice cream is the most delicious.
Use the verb disagree to talk about what happens when you have a conflict or a difference of thought about something with another person. People disagree about politics, food, sports, and movies — there is no subject that won't cause most people to find a way to disagree. The word comes from combining the Old French agreer, "to receive with favor or take pleasure in" with the Latin prefix dis, which here means "do the opposite of."