My understanding is that the early Christian church adopted the canon of Hellenistic Judaism, commonly called the Septuagint. This was a Greek translation of the Jewish Canon, and included books that were not found in the Hebrew version.
However, the early Church distinguished between the books common to both the Hebrew and Greek Jewish scriptures, and the books derived solely from the Septuagint. The books common to both traditions became the Old Testament. The books found only in the Septuagint are called the Apocrypha.
(Chaim - have I got the references to the Septuagint right?)
The early Church had different books floating around, so to speak, with some accepted unversally, and some only regionally. A Council at Rome, convened by Pope Damasus in 382, established the canon of books of the OT, Apocrypha, and NT, for the Roman Catholic and Greek Orthodox churches.
During the Reformation, some Protestant leaders refused to accept the canonical status of the Apocrypha, since those books were not found in the accepted Hebrew Canon. This became a point of diversion between the Roman Catholic and the Protestant churches, as the Roman Catholic Church re-affirmed the traditional canon at the Council of Trent, one of the major stages in the Counter-Reformation.
Martin Luther did not get rid of the Apocrypha entirely, unlike some of the Protestant leaders - he included it as an appendix to his Bible. As well, in the Anglican communion, some of the books of the Apocrypha are used as part of the accepted lectionary, but are not canonical - this is a point of distinction between the Anglican communion and churches such as the Presbyterian church.
I believe that to be accepted into the New Testament, there were two requirements: the Council held at Rome in 382 had to conclude that:
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the book in question was a true source of spritual inspiration/Christian doctrine, and,
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its authorship could be traced to an Apostle, or an associate of an apostle.
The second criterion was a way to ensure authenticity. For example, the Gospels of John and Mark are traditionally ascribed to theose two apostles, while Matthew was considered an associate of Peter, and Luke of course an assocate of Paul. (Modern textual criticism has cast some doubt on these traditional ascriptions, I believe.) By this criterion, the category of books eligible to be considered for the Canon closed when the initial generation of apostles and other disciples died out.
To further confuse the situation, there are also the Apocryphal New Testament books. These are books written around the same time as the accepted NT books, but which were rejected from the Canon, either on grounds that they did not represent true doctrine, could not be traced to a particular apostle, or both. Some of these are pure miracle stories, others are collections of gnomic sayings. One of the most interesting of these is the Gospel of Thomas, a collection of sayings attributed to Jesus. Some of those sayings are found in the canonical Gospels, but others are found solely in Thomas. Of these, there are some that NT scholars think may truly represent additional sayings of Jesus.