This writeup will cover not just the Battle Royale book, but also the manga, since information on both seem to be somewhat scarce at the moment. I'll also include a few points the various incarnations of the story differ from each other.
Battle Royale book
Author: Koushun Takami
Publisher: VIZ, LLC (published in Japan by Ota Shuppan)
ISBN: US edition: 1-56931-X
Koushun Takami was born in 1969 in Amagasaki, near Osaka, and grew up in the Kagawa prefecture of Shikoku. He graduated from Osaka University with a degree in literature and later dropped out of a liberal arts correspondence-course program. He worked for the Shikoku Shimbun news company from 1991 to 1996, and shortly afterwards wrote his first book, Battle Royale, which caused a great deal of controversy in Japan.
In the book, Japan did not lose World War II and has developed into the Republic of Greater East Asia (i.e., Japan and presumably a number of colonies). It is isolated, somewhat united in its opposition to the American Imperialists and is ruled by a Soviet-style government, complete with a leader aptly known as the Dictator.
One of the defining characteristics of this republic is the Program. The Program is a yearly event where each prefecture randomly chooses a junior high school class, which is then transported to a secluded area and told that they will have to kill each other. The last one standing after 24 hours is the winner. Should there be more than one left, then the collar each student wears around his/her neck will explode and there will be no winners. The aim of the Program seems to be to encourage young people enlist, since Japan (unusually for a dictatorship) does not have a system of conscription. The exact logic behind this is not readily appearent, though.
The book details how each member of the Shiroiwa Junior High School's third year class B handles the situation when they are confronted with this choice. While the main character, Shuya Nanahara, is followed most closely, the fates of all his classmates are included. Some go into denial and are inevitably killed by the more enterprising students who have decided to play the game, while others attempt to cooperate and somehow defy those who have forced them into an intolerable situation (with varying results).
While the book can be depressing at times, it becomes clear that every student has a reason for acting like they do (something which is not always obvious in the movie). Throughout the book, the protagonist and his friends sometimes discuss the nature of the society they live in, the effects of the repressive dictatorship and what drives their class mates to, in some cases, cheerfully kill their former friends.
While I haven't read the Japanese version, the translation, done by Yuki Oniki, seems to be done fairly well. However, there are a few points where it seems like a passage has been translated directly from Japanese. These passages seem out of place, and should, perhaps, have been rewritten to suit the style of the rest of the book.
Battle Royale manga
Created by Koushun Takami and Masayuki Taguchi
ISBN: US edition: 1591824345 (first volume)
The Battle Royale manga is based on the book with the same name, and the plot is (at least during the first two volumes) nearly identical (see above). Compared to the book, the manga is gratuitously bloody. Not even the movie manages to portray the slaughter so vividly. While the book sometimes describes a corpse or somebody's death in detail, the manga shows an abundance of blood, brains and gore. While not exactly offensive, it does tend to detract from the plot when the reader is repeatedly treated to close-ups of spectacularly maimed corpses. The art looks pretty much like you would expect a Japanese comic to look, although each and every character seems to have a body which mirrors their personality: One look at somebody's face, and it becomes annoyingly obvious what we can expect in the way of action from him/her.
At this point, at least seven volumes have been released in the US, and there are probably more coming.