An Heir can be one of your children or other living NPC within your family and will inherit your possessions, land, title and buildings within Elyria when your character dies.
As you should already know by now, within Chronicles of Elyria your characters ages, and eventually, whether killed in combat or from old age will die. If you've built up a known family name, hold a title and/or own land or properties, then you'll most likely want to continue owning them with your next character. - this is possible with an heir.
As long as your character has gained enough Story Points during their life (by completing quests etc), then when they die, you'll be able to choose to live on using an eligible member of your dynasty, or a child.