To disown someone is to reject them. If you disown your brother, you refuse to have anything to do with him: not only do you not speak or have contact, but it's as if he's no longer related to you.
When one person disowns another, it's because of some terrible argument or deep-rooted conflict. Although it's uncommon to disown another person, when it happens it's usually a family member who's cast off. Your mom might threaten to disown you after you drive her car into the mailbox, but she's probably not serious. Disown takes the root word own, meaning "to have or to hold," and adds the Latin prefix dis, "not" or "do the opposite of."