Ai Yori Aoshi is a manga which spawned two anime series and four "visual novel" style video games. The name could be translated as "Bluer than Indigo" from the Chinese phrase Qīng chū yú lán ér shèng yú lán which when translated to Japanese becomes Ao wa ai yori idete ai yori aoshi, "Blue comes from indigo, yet it is bluer than Indigo." This phrase reflects on the idea that although people are shaped by their teachers, they later surpass their teachers. There are at least two puns in this, the main female lead is named Aoi ("blue") and it is pronounced very similarily to Ai ("love").
This series is often compared to Love Hina.
The story of the first series of Ai Yori Aoshi revolves around a university student named Kaoru Hanabishi, who appears to be just a normal student at first, but it turns out he is the son of the leader of the Hanabishi zaibatsu. When Kaoru was five, his father died, and his grandfather took over his education, grooming him to be the next leader of the zaibatsu. Kaoru never felt comfortable with the Hanabishi family, and following the death of his mother, he leaves to live on his own.
The female lead of the series is Aoi Sakuraba, whose parents own Sakuraba Department Store. Unbeknownst to Kaoru (but knownst to us1), Aoi is deeply in love with Kaoru, and in fact, both of their families had agreed on marrying the two of them. When Kaoru left home, however, the marriage was cancelled. Aoi leaves home as well, to chase after Kaoru.
Both of them, now free of their family affairs, find that they don't know how to make their own livings. Miyabi Kagurazaki, who keeps an eye on Aoi, comes up with a plan to let them be together. Miyabi moves Aoi into a large summer home owned by the Sakuraba family, and lets Kaoru stay in the servant house nearby, thus preventing the scandal that would result if anyone knew they were living together. But, shortly after Kaoru moves in, his friends come to live at the home as well. And since most of Kaoru's friends are beautiful young female students, the house quickly turns into a girls dormitory, with Aoi as the landlady.
The events covered in the first series take place over the course of a year or two, and the second series, Ai Yori Aoshi:Enishi, takes place about a year after the end of the first series. In Enishi Kaoru is now a grad student, and the relationship between him and Aoi is developed.
1. Sorry, couldn't resist the Spaceballs reference.
Thanks go to Raincomplex for pointing out a correction.