I just watched the whole series (1-13 episodes the OVA) today, the anime from episode one hooks you until the very end, I wasn't expecting the storyline to be as compelling or as emotional as it was but this wasn't the first anime that surprised me this way and i'm sure it wont be the last.
Lucy is a Diclonius (two - horned human with telekinetic arms called vectors) hell bent on destroying man-kind to create the next race. Nyu is Lucy's alter-ego / personality whatever you wish to call it, if anyone has seen Chobits just picture Chii and you'll pretty much have a perfect match to Nyu.
Without going into too much detail and spoiling the anime for you, I will offer a little warning this anime isn't for the faint of heart / stomach or children under 16.
There is a lot of nudity, blood, guts and violence, and should be watched at your own risk don't say I didn't warn you.
I hope you enjoy Elfen Lied as much as I did and I hope you find this summary helpful.