Now it seems as if many of you who have watched Fooly Cooly actually just enjoyed the random hilarity of the series but failed to understand the plot. Sit back, and relax. I'll try to explain it. Chronologically.
Atomsk, the Pirate King, is the most powerful space pirate in the universe. It is unclear what relations, if any, Haruko had with Atomsk, but one thing is for sure: She wants his powers. She's got a bracelet that looks the same as Atomsk's nose ring, and it shakes when they are close together, which allows Haruko to notice if N.O. channels are opening.
Atomsk is captured by Medical Mechanica. Who they are is not exactly explained, but we know one thing: they are a monolithic industrial corporation that makes a lot of robots.
Haruko is now determined to bring out Atomsk from Medical Mechanica's confinements by directly warping robots randomly out of N.O. channels. To do this she needs brains that matches the proper signals, so she comes to Earth in hopes of finding someone.
At this time, Amarao is a child who Haruko first choses. She goes by the name Raharu (its either a nickname or her true name, Haruko could also be an alias). Unfortunately, Amarao's N.O. channel is weak, so he cannot pull much out of his head. Haruko leaves.
Medical Mechanica reaches planet Earth. Amarao becomes commander of the Interstellar Immigration, in order to negotiate with Medical Mechanica. Things look bad for Earth, as Amarao is fully convinced that MM is hostile and they are barely holding back from taking over. A giant Iron (a medical mechanica plant) is placed in Naota's town (thus Naota's explanation, "the adults were excited when the plant came in, refers to their anxiety with the aliens. Naota is too young to understand this). The plant, according to Amarao, is a weapon that threatens Earth.
Naota is a lonely boy who tries very hard to be as adult as possible. The adults around him, however, act very immature. His brother, Tasuku, is the very epitome of being an adult for Naota, and so he tries his best to mimic hm in every way possible. This also explains why he says "nothing ever happens" even with all the chaos, because he believes being calm is the same as maturity. Its also the reason for his fixation with sour drinks, even though he hates sour stuff.
Haruko once again arrives in town, and this time she has Naota's head as her target. Unlike Amarao, Naota's head is the "right kind of head" for N.O. channels, and proves to be very effective in pulling out lots of MM robots. Haruko takes residence in Naota's house and constantly provokes him to bring out more N.O.
N.O. channels turn the user's brain into mush, and creates horns which slowly come out as the robots. The first batch brings out Canti. He was taken over by Atomsk as a means to escape, but after taking a blow in the head by Haruko (unknown to her knowledge), Atomsk turns dormant inside of Canti.
Atomsk's powers will rise again when Naota is connected to Canti. This first happens during the fire starter incident.
Ninamori is the first person besides Naota to open an N.O. channel. Her head was also the "right type", but wasn't activated until her head collided with Naota. Having seen this, Haruko begins to experiment on Naota's father head to see if a channel could be opened. This is the scene where Naota accidentally believes Haruko and his father are having sex. His head fails, and Haruko stuffs him in a closet and creates a replacement robot.
Amarao first contacts Naota. He explains what Haruko is up to, but his understanding is not actually clear either. He believes Haruko is lovers with Atomsk. He understands that continuously releasing MM robots is not good for negotiations and wants to put an end to this issue as fast as possible.
Naota "swings the bat", saving the city from a MM bomb. This is the first time he's faced his fears of not being able to match up to his brother. Mamimi loses interest in Naota around this point, since she has no care for those who are self dependent.
Amarao already knows that Canti could be Atomsk, so he sends Lieutenant Kitsurubami to assassinate Canti, but she fails.
The final MM robot appears and tries to activate the plant. Canti pulls out Atomsk's signature weapon, 1961 Gibson EB-0, fully confirming that Canti houses Atomsk's power. Canti defeats the MM robot, but its core manages to escape by itself. Haruha goes into hiding because she's being chased down by the Interstellar Immigration.
Since Mamimi is now disinterested in Naota, and her cat has found a partner, she is left with no one worse off than her. She finds the MM robot core, and decides to take care of it as a replacement for both Ta-kuns.
Amarao tries to patch up Naota's head, so he suggests he drink sweets (glucose recovers the brain). He sticks seaweed on his head, as it blocks N.O. signals (also why he wears them too). Mamimi begins to feed the core to exact revenge on the classmates and townsfolk who made her miserable, and it begins to grow.
Canti was originally made to be a part of the terminal core, so he merges with the core and activates the plant. Haruko uses Naota as a catalyst, and the machine is fully activated. Right before the machine activates the plant, Naota comes back out, having completely swallowed Atomsk's powers.
Haruko's entire plan was to have MM destroy Earth, while she extracts Atomsk's power for herself. Now that Naota has the power, she goes into a rage and attacks. After the brief fight, Naota confesses his love for Haruko, and releases Atomsk through his N.O. channel.
Atomsk destroys all MM robots, and escapes. Haruko offers Naota to come along on the chase, but leaves him until he is ready.
"Her target is an astral being named Atomsk. They say he has so much N.O. that he can steal whole star systems. Normally he and Raharu would be mortal enemies."
The plot of FLCL is really interesting. Try watching it again with this understanding, and you'll see it in a completely new way.
The manga shows Haruko's darker personality more. She even tells Naota that she no longer needs him because she found other heads that can use N.O. channels, and even tries to kill him. In both the anime and the manga, she shows no interest in the safety of the humans, saying she doesn't care about them as long as she gets what she wants.
Leaving Naota, she takes his (better) guitars, and leaves her old ones. Overall, she's a douchebag.