In Hallfreder Saga, as he lay dying, Hallfred saw a woman in armour approaching him. He called her his fylgjukona.
The fylgjur is rather like the banshee, a protective spirit that is rarely seen by the people whom it protects except when someone is about to die. The banshee wails in mourning as a harbinger of death, not as a bringer of death. Banshee simply means 'woman of the Sidhe' or fairy woman.
Fylgjur may be related to fylgja or afterbirth. The belief was that when a person was born, the afterbirth followed and that was the person's fylgja. You had to be careful in disposing of the afterbirth least an animal consume it and the person's soul with it. It is interesting that Icelandic peoples thought the afterbirth was an Otherworldy being. Also interesting that Celts and Norse thought this spirit was a woman.
(from Cassell's Dictionary of Norse Myth and Legend by Andy Orchard)