A Series bible is a guidebook written by the creator(s) of a TV show so that all writers will know the general outline of the show; it contains the backstories of the characters and outlines the in-series universe. Typically, it contains information that might not actually be revealed to the audience until later, but is known to the writers from the start, so they won't make any conflicting new stories.
The Battlestar Galactica Series Bible was written by Ron D. Moore after the Mini-series during the break before production on the regular series began. It has not been released in its entirety, nor will it probably, because it still contains much information not revealed on screen yet. However, the series bible is not necessarily definative, because sometimes the writing team decides they really want to do something differently than was originally outlined. Still, so long as nothing new is made that conflicts with it, it contains the real backstories for the characters and basic plot outline for the future of the show.
For example, RDM has said in his season 1 podcast that according to the series bible Starbuck used to be a Pyramid player and was going to play professionally but got a knee injury and had to drop out and find a new direction for her life. Several episodes later in season 2's "Resistance" Starbuck states on screen that she used to play pyramid but a knee injury forced her to stop.
RDM often refers to information from the series bible in the podcasts (usually, things he wanted to put onscreen but could not). Another example is Col. Saul Tigh; RDM has stated that his series bible entry says he was a young Petty Officer not unlike Chief Tyrol when during the First Cylon War his ship, Brenik, was boarded and the fighting with Centurions was some of the bloodiest of the war, devolving into hand to hand combat. Later, Tigh was stationed on another ship which was also lost, and then he was dragooned into officer school to become a Viper pilot when Colonial numbers started running low. The series bible also fully described his poor relationship with his wife, Ellen Tigh. Originally, all the viewers would ever see of this relationship was that it (combined with Tigh's horrific war experience) had driven Tigh to alcholism as seen in the beginning of the Mini-series, also when Tigh is burning her face out of a photo. This was supposed to be all we actually saw of Ellen Tigh. However, the other writers read the full description of her in the series bible, and subsequently thought she's make a good character, and thus "resurrected" her in "Tigh Me Up, Tigh Me Down".
There is still much material from the Series bible which has yet to play itself out on the show, but may be gradually revealed.