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Also called "Fooly Cooly," or in Japan, "Furi Kuri," which is apparently slang for "fondling a girl" (or fondling breasts, or something like that, I dunno). Um... so there's this kid named Naota whose older brother moved to America to play baseball, and now his girlfriend Mamimi acts like she's interested in Naota. She calls him Ta-kun, which other people in the anime apparently pick up on and start doing, though Naota doesn't seem to like it much. Mamimi also called a stray cat Ta-kun, and I guess she used to call Naota's brother that, too. She's a fairly weird girl, but not nearly as weird as this other girl named Haruko, who drives a Vespa, and is apparently an alien.
Strange stuff starts happening after Haruko shows up, like things coming out of Naota's head. Apparently she turned his head into some kind of interdimensional portal. The first thing to come out of his head was a robot, who Mamimi started calling "Canti" (or "Kanchi" in the Japanese version), after a god in a video game she played all the time. Canti now stays with Naota's family and does chores for them. Haruko is also staying with Naota's family, ostensibly as their housekeeper, I guess. She had a hidden agenda, of course, but I have no idea how to explain it. Anyway, the whole series is kind of weird, intentionally so, mixing in animated manga and stuff, and characters go spastic a lot. I'm not sure what else to say. I couldn't follow the plot very well, but at least it wasn't boring or anything, and it was only 6 episodes. Also it had good music, by the Pillows. Someday I'll probably rewatch the series and possibly write a better review....
In 2018, there were a couple of sequel TV series, "FLCL Progressive" and "FLCL Alternative," but I'm not sure if or when I'll get a chance to see them.