Well, Elfen Lied is basically about an OP psycho girl. She has horns and 4 extra invisible arms that grow out of her back that can pass through anything. You've watched the first episode, her powers are well demonstrated there by killing many people. Many will say those people are not innocent, some will say they deserved it, but I disagree.
Then she crashes, naked, at some "random" dude's home, accompanied by his cousin (whom many fans of the show hate because she gets jealous a lot, but I think she's underrated and is one the nicer characters of the show). However, she is nothing like the girl who only recently murdered many people, she is now barely short of a lovable and stupid puppy.
The plot "progresses", the facility she was kept at sends two supporting characters to capture her. One is this buff military soldier dude who gets badly injured, whom Lucy (the psycho chick) could have easily killed but did not (which is very odd considering she did not hesitate to kill anyone else), the second is person like Lucy (Nana, the only character I sincerely liked), and while Lucy could have easily killed her, Lucy settles for amputating her four limbs. Why she did not settle to kill two people who were after her and decided to kill all the other people beyond me.
Eventually, Nana comes to live with them and is joined by a not very important character who was a victim of child abuse..
Eventually, we find out Lucy is the childhood friend of the "random" dude she crashes with, whose his sister she killed along with many other people, including more children, as a child "incapable" of controlling her own powers.
Shit happens, Lucy fights the organization that kept her and used her for experiments, undoubtedly killing more people, and it becomes ambiguous to whether she survives or not. However, in last few scenes, someone rings the door bell to random dude's home and it's most likely Lucy.
I had several problems with the show. One was the plot holes, including that of which Lucy coincidentally did not kill several people who were of importance to the plot when she easily could have. Another was the fact that the anime portrayed Lucy as an "antihero" when in fact she's either mentally ill, the villain of the story, or both. What Lucy did was unforgivable, and on top of it, she kisses the random dude, all the proof I need she was not even sorry for what she did (a sorry person would have thought they didn't deserve love or anything similar). I've come across people who talked about "redemption," but Lucy did not even redeem herself.
This is why Elfen Lied is an abomination. It is an anime that tries to justify pointless killing. Some of the people Lucy killed may have not been entirely innocent, but as a result of their deaths, their families will grieve. On top of that, it was not even out of self defense. She killed them in cold blood.