The Tale of Genji, or Genji monogatari, was written in the tenth century by Shikibu Murasaki. In it, there is a deep look at the culture and way of life of the highest classes in Heian-era Japan.
The tale itself is about the 'shining prince'; Genji. Son of the emperor and one of his lowest consorts, Genji is fated to be one of the most important men of the age, but never able to truly ascend to royalty. This story, all thousand plus pages of it, details his life full of music, poetry, and efforts to win the hand of various ladies.
While starting out very episodic, Genji soon turns into a more refined tale, when all the threads of story come together to create surprising relations and events that will delight the imagination in their color and depth. By the end of the book, you will have lived through so much of the characters lives that each person comes into their own, and you cannot help but hope that all will end well.
I will say, however, that this book is somewhat difficult to get into for the uninitiated. There's much in the way of allusion to religion of the day (be it Shinto or Buddhism), and of customs that are barely mentioned due to being so commonplace at the time. As such, I would suggest something to introduce people to the Heian culture.
My first and best suggestion would be The Tale of Murasaki, written by Liza Dalby. It's a diary of the author of Genji, Shikibu Murasaki, pieced together from poems and the real diary, and filled in with further guesses as to her life. Compared to Genji, it is very approachable, and makes reading this story even easier.
I cannot recommend Genji enough, being quite possibly the first novel in the world, and certainly one of the best. For anyone with an interest in Japanese history, well-written romance, or just the best of the written word, Genji is sure to delight.