First of all the concept of “The Bible” is fairly modern – in ancient times, the Scriptures were a collection of separate books (actually scrolls.)
In the time of Christ, the Saducees accepted only the first five books of the Old Testament (the Pentateuch or Torah) as Scripture. Other Jewish sects accepted more books – perhaps some that vanished from history and were only found in the Deat Sea Scrolls.
The Ptolomies of Egypt were all patrons of the Library of Alexandria. Ptolomy also encouraged Jews to settle in Alexandria – and asked them for a copy of the Bible – in Greek – for the Library. The Jewish leaders in Alexandria were initially reluctant to translate the Bible, but came under pressure from the Jewish people in Alexandria, who spoke Greek, not Hebrew as their “mother tongue.”
The Bible (or most of it) was translated by a group of 70 Jewish scholars (some sources say 72) and this Greek translation is called the Septaugent (sometimes abbreviated LXX.) By about 200 BC, all of the current books of the Old Testamen had been translated and were considered a definitive list of the inspired works by most Jews world-wide.
The Septaugent was accepted as valid even in Judea, where the people spoke Aramaic, not Hebrew anymore. Most educated Jews spoke Greek, however, and the Septaugent was the most widely-used Bible in Jesus’ time. Note that in the Gospels all Jesus’ quotes from the Bible are from the Septaugent, with its characteristic phrasing.
The Church slowly accepted the idea of a New Testament, as well, and finally Pope Damasus I proclaimed the Canon of both the New and Old Testaments near the end of the 4th Century. The Jewish Canon only evolved a few hundred years later, under the leadership of the Ben Asher clan, the Masorites (“handers-down.”)
For more than 1600 years, from the time of Christ until the Reformation, the books listed by Damasus I were accepted thorughout Chrisendom.