When two stubborn people are arguing about something, they're at loggerheads. If you want pizza for dinner and your sister wants falafel and you both refuse to compromise, you're at loggerheads.
When a family's at loggerheads, no one agrees with each other, and everyone has strong opinions. Whether you're at loggerheads over what movie to watch or who gets to sit in the front seat, it's a frustrating situation. This phrase stems from the old-fashioned insult loggerhead, "stupid person or blockhead." It was also the name of a heavy iron tool, and the "battling" sense may stem from the idea of swinging big, blunt weapons at each other.