FLCL is simply the best anime ever. A 6 episode series, it's less than 3 hours long in total, shorter than many movies. I've read a review stating that watching FLCL is like falling in love for the first time again, and it's a great way to put it. In summary, FLCL is a story about a 11 (I believe, may be 10) year old boy named Naota and his brother's strange girlfried named Mamimi, and what happens when they meet a strange 20-year old woman named Haruhara Haruko who claims to be an alien.
The first 2 episodes will confuse you, but will leave you curious. Episode 3 is the weakest, and may draw some people away. Episodes 4-6 are simply incredible, as we find out just what's up with Haruko and why it matters. The climax in episode 6 is simply beautiful, stunning, moving.
As many people have said, FLCL is very confusing. The confusion is not an especially frustrating type, because we understand what happened, but not always why. But in the end, it's easy to see that it's more of a coming-of-age story than anything. Some have raved about the soundtrack, and it indeed is great, the best in anime. Most of the background music are rock songs by a band called The Pillows, and they are simply perfect for every scene they are in. Every episode but ep3 has at least 2 scenes when I was simply in awe of how well music could fit with the scene.
When I think of FLCL, I think of it more as a beautiful piece of art that used anime as its medium, not the other way around. FLCL doesn't really fit in any of the classic genres of anime, although the plot may. I encourage anyone and everyone to see it in its full.