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 re-imagined Battlestar Galactica series bible was written by Ron D. Moore after the Mini-Series during a hiatus before production on the regular series began. It has not been released in its entirety, nor will it probably, as it still contains much information not yet revealed on screen. However, the series bible is not necessarily definitive, as sometimes the writing team may decide something different must be done in a script than was originally outlined. Still, so long as nothing new is made that conflicts with the bible, 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, intending 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 "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 Colonel Saul Tigh; RDM has stated that his series bible entry says he was a young Petty Officer not unlike Chief Tyrol when during the Cylon War his ship, Brenik, was boarded and the fighting with Cylons 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 alcoholism as seen in the beginning of the Mini-Series, also when Tigh is burning her face out of a photo. Orignally, this was supposed to be all viewers actually saw of Ellen Tigh. However, other writers read the full description of her in the series bible, and subsequently thought she'd make a good character, and thus "resurrected" her in "Tigh Me Up, Tigh Me Down".