The process of working out which books are 'in' and which are 'out' was a long process, and it occurred differently for the Old Testament and New Testament.

The Old Testament in its standard Hebrew Bible version was formalised sometime in the late 1st century. It used to be held that this occured as a council at Jamnia ca. 90 AD, but this is now regarded as a hypothetical construct.

However, it seems that a fairly recognisable collection of Scriptures existed before this. That is, the first century gives you a date at which things are settling, but things got 'settled' before then. Josephus, ben Sirach, and even the Qumram collection all attest that there was something like a 'working canon' at this time.

This is complicated when you begin to discuss the Septuagint, the Greek translation of the Old Testament, which contains several books not found in the Hebrew Bible and not generally accepted in Jewish circles. However, this was the version of the Old Testament widely available through the Greek speaking world, and used almost universally by the early church, only really contested by Jerome.

The New Testament canon emerges over a period of some hundred years. Books that emerge in the 1st or 2nd century that were of similar genres (gospels, apocalypses, epistles) that were not accepted into the NT Canon are generally called 'Apocrypha', and you can get a list of them easily from wikipedia, and many of them can be read in translation online.

The first traces of 'what is authoritative' once you get past the NT period is simply observing what early christian writers use - who do they quote, who do they show familiarity with, what authorities do they call upon.

I think there's good reason to see the pressures of the 2nd century pushing early christians to define more carefully which texts they considered 'authoritative', under the influence of Gnostics and followers of Marcion. Gnostics, although even the label has come under question recently, is an umbrella term for various movements that had some common interpretive paradigms that were relatively divergent from the Christianity that emerged as 'mainstream'. Anyway, Gnosticism produced its own literature that claimed authority. In the same time period Marcion was rejecting most of the New Testament books and using a canon that was basically edited versions of Paul and a streamlined version of Luke.

You see actual 'list-making' of accepted and rejected books emerge with the "Muratorian Canon", a list that is associated with the church in Rome, and dated ca. 200. Here is an online translation of the text. You get further lists in Eusebius' Ecclesiastical History, which lists books as 'universally accepted', disputed, and outright rejected, and then further lists through the 4th century in particular. In Athanasius' 39th Paschal Letter there is a list pretty identical to the NT as accepted in Roman Catholic, Protestant, and Orthodox churches. The Church of the East and Oriental Orthodox churches have a slightly different New Testament Canon.

I think I will stop there for the moment. I can answer follow up questions though.