You will need the Limited Write People permission to perform this action, at one or more of the family's campuses. It is campus-scoped, so a campus override can change it at one campus. See People for what the permission covers, and Campuses and organizations in Version 8 for how campus access works.
To remove a family member
To delete a family member, select the pencil icon beside the Family column on the right-hand side of that page > choose the 3-dots menu beside the person to be removed and select Remove from family.
To remove a relationship
Select the tab labeled Relationships, click the 3-dots menu and choose Remove. The Relationships tab is for external relationships, which can include grandparents, cousins, a separated or divorced parent, etc.
Helpful Tips
This only removes the person from the family and relationship area. It will not delete their profile from the system.
A family manager cannot be removed from a family. First, someone else would need to be made family manager and then the person could be removed.
Family members and relationships at campuses you cannot reach are left out of the family panel and the Relationships tab, so you will not see them to remove.