In the first episode of Elfen Lied, Lucy is pretty naked through the whole episode until she meets Kouta and Yuka at a beach. Of course in any anime I've watched there is always someone who loses their memory or can't talk and Lucy is both. Since she can't talk and the thing she says is "Nyuu", Kouta and Yuka names her "Nyuu".
As the story progresses they find out that Nyuu is wanted by the government, but they don't find out why until the near end of the show: Nyuu real name is Lucy and she is a killing machine called a dicolosis (I might have spelled it wrong).Mayu, a run away girl who joins the family, is the first to have a suspicion about Nyuu.
A bunch of other dicolosis are sent to find and destroy Lucy and also end up being the daughters of the guy who started the project to seal the dicolosis' away from the world and then use them as super weapons. I would tell you more, but that would give you hints about the ending.