Before Jesus was born, the various scrolls comprising the Hebrew scriptures were in wide use and that was the part of the Bible Jesus and the apostles used. As it forms about two-thirds of the Bible, it's important to know its story as well.
Flavius Josephus, the famous Jewish historian, wrote 'Contra Apionem' dated around AD 100, and quotes parts related to the Hebrew canon. For him the canon was closed, and had been from the time of Artaxerxes, 465-425 BC - essentially, the time of Malachi.
The Dead Sea Scrolls had quotations from most of the books of the OT, showing all of them to be canonical no later than 130 B.C. By the time Jesus was born, those books were collated and used in the synagogues. There were 7 additional books about the history of the Jews during that 400 year interval which Catholics included (and called them 'Deuterocanonicals' but which Protestants called "The Apocrypha"). Now for the Christian Greek scriptures.
Jesus had lived and died by around AD 30. Writings about him began at the earliest in the AD 50s, some 20 years after his death, and all but the last letter of the New Testament was finished by 75. The last book that comprises the NT was finished by about 95 at the very latest (though some argue for an earlier date). The critically important point is that all the NT books were written by people who lived with and knew Jesus, or through the accounts of eye-witnesses told to them. Paul encountered the risen Christ on the Road to Damascus and his life was transformed by that event, and he was accepted as one of the apostles. All of the Christians of the first century AD had those various letters circulated and copied to them and knew them all. Some additional writings were dismissed by the early Church Fathers as Gnostic heresy, or as not inspired.
Two Church Councils (Hippo, AD 393 & Carthage, 397) had a canon of 46 books for the OT and 27 for the NT - just what the later Protestants had in 1529, though the 7 Deuterocanonicals were put by Luther into an Appendix at the end of the Bible. The settled canon was exactly the same back then as for the much later Council of Trent in 1563. From around 1826 the Protestants didn't include the Apocrypha, even as an Appendix. But the Council of Nicaea in 325 had nothing to do with the canon of scripture.
Before the Reformation, the Western Church used the Vulgate Bible, which was in Latin. It was a closed book to most of the populace. An 'infallible' test for revealing a heretic to Roman Catholic authorities in the Middle Ages was to see if they possessed, or even knew any part of the Bible in their own language.
Guttenberg's first printing job was the Bible, in 1453. It was not the Vulgate. This one had been translated from the original Hebrew, Greek and Aramaic, known as the Majority or Traditional Text. Erasmus published the first printed Greek New Testament in 1516, then came the Tyndale/Coverdale Bibles in 1525; the Geneva Bible in 1560; the Bishops' Bible in 1568 and then King James I of England & VI of Scotland commissioned the KJV which was published in 1611.
The KJV had used the Hebrew Masoretic Text and the Greek Received Text alongside working from the Bishops' Bible. You can get a list of the various translators at http://www.jesus-is-lord.com/transtoc.htm