You will need the Limited Write People permission to perform this action.
To remove a family, navigate to the profile of any member of the family and click on the "Family" tab. Next, select "Delete family" from the drop down next to the family.
You cannot delete a family marked as primary. You will first need to mark another family as primary before removing that one.
Deleting a family does not delete any of the members of that family from the system. Instead, each member of the deleted family will be made the head of household of a new family. They would be the only member of that new family. The new family would become their primary.