The Mother of the Fog is a myth in Fallout 4.
The Mother is a sacred spirit worshiped by members of the Children of Atom, as they believe her to be a saint of their god, Atom. The player finds her when they drink irradiated water from Atom's Spring in Far Harbor during the quest Visions of the Fog. She is a dark, ghostly figured shrowded in black smoke. She leads the player to an abandoned building in the middle of the fog. It is here that they will find the "mother icon", a small statue.
The Mother is not seen again after the quest but many players claim to have seen her appear then disappear in various locations in the fog of Far Harbor.