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What is a Diclonious?
As far as I know a Diclonious is a type of mutated human or something. They have horns on their, head and the only ones we've seen so far are female. And they have pink hair!!!
Although there were two males in the anime, or so I've heard. They both died I think. I can't remember...
Anyways by the time a Diclonious is 3 it can use its vectors and usually kill the parents. Although theres usually one or a few humans, that a Diclonious could never kill. Sometimes a Diclonious will go temporarily insane and kill humans, oddly enough every Diclonious victim has been human. Sometimes when a Diclonious is born, its killed before it leaves the nursery...
Diclonious siplets (Parents were infected) are unable to bear any child so they exist to destroy, or thats what someone said in the anime. Sometimes a Diclonious will develop a nature where they never want to kill...
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What are vectors?
Vectors are a type of Telekinesis where a Diclonious has these 'hands' that have a limited range and a limited amount of 'hands.'
There highly dangerous as they can tear off someone's limbs in seconds, it can even block bullets and pick things up and use them as projectiles. Although it does have a weakness...it can't block armor piercing bullets...

Character bios 2: Kouta

Kouta is the main dude in the anime. He is rather friendly and energetic. He moved to another town to study, but everything turns to a chaos after he meets Lucy who he refers to as Nyu, at the time Nyu had been shot in the head and her personality split in two. Nyu at his first day at his new accomodation. Nyu is alway in trouble, therefore Kouta gets very concerned about her as the nice guy he is.
And without even knowing this behavior hurts Yuka, thus causing all three a hard time.

Kouta was Lucy's first real friend when they were kids, at one time Lucy thought about killing him when Kouta first saw her horns, but she changed her mind when Kouta said they were cool...
Kouta liked to hang around with Lucy even though they were together for only a short time.
Kouta's biggest mistake was probably saying his cousin (Yuka) was a boy. So he had lied to Lucy, probably because he didn't want to make her feel bad because he could tell how she felt.

He's usually kind to everyone he meets, though the first time he had met Nana he slapped her for attacking Nyu. Eventually he forgives Nana and lets her stay at the Maple Inn. He wants to know why Nyu has horns, though during the last episode he remembers who Nyu is and it's the 2nd time he comes into contact with Lucy after 8 years. Actually 4th if you count episode 6 and 8. I just count it as the 2nd because at that time he sees Lucy kill again, and at that time he's actually aware that it's not the same Nyu he knows...

(Major Spoiler: Don't read on if you don't want the ending to be ruined XP)

During the end of episode 13, when Lucy is about to leave Kouta stops her from leaving. At first Lucy is shocked having killed Kouta's father and sister. But Kouta admits he can never forgive Lucy, Lucy thinks that Kouta hates her, but she's quickly mistaken when Kouta admits that he loves both Lucy and Nyu, because he sees them as one person.
After that if you've only seen that part you'd think that Lucy and Kouta end up together for the rest of there lives, but the people who created Elfen Lied decided to add another unanswerable plot twist where while Lucy is heading home she is cornered by soldiers and than they shoot at her, we don't know exactly what happens because the camera quickly shifts into the sky and we sww two lamp posts being bent...than a few seconds later we see Lucy's remaining horn broken off and fly into the air with blood stains...At first we think Lucy dies, but than later at the end someone appears at the gate to the Maple Inn, although the door never opens for us to see who it is but we all think(Know) it's gotta be Lucy...Who else could it be?