The technical name for this process is Canonisation, i.e. deciding which books are part of the Canon, or Rule, of Scripture.

It was a much more complex process than your question text seems to suggest, and a lot of that process is unclear to us because we simply don't have evidence on it.

You need to remember that there wasn't just a bunch of books and some big-wigs decided these ones are in and those ones are out.

No, instead you have all sorts of books, and you have early communities and their leaders using them, reading them in meetings, appealing to their authority, and quoting them in their own writings. And different communities are using different books. The process of canonisation really gets into swing through the mid to late 2nd century, as Christian writers begin to talk about a 'New Testament', and there is an overall general agreement about what books will be in, but some books are disputed. In particular, for the NT, the books of Revelation, Hebrews, James, Jude, and 2 Peter continued to be debated.

The process then was largely one of emerging consensus about which books could reasonably claim to have emerged from the Apostles, and accorded with the doctrines of the Church, and appeared to address the church in general. There was not really a point where someone sat down and said "okay, we got to get rid of this book".

We have a number of texts that list of the books of the NT, generally starting with the Muratorian Canon usually dated to ca. 170, through Origen, Eusebius, Athanasius. Eusebius' list from the 4th century divides books into accepted/accepted but disputed/not accepted/heretical.

It's worth stating that 'the Bible' is not the same across all 5 major Christian groupings. Especially with regards to the Old Testament. Wikipedia actually has a very nice chart depicting this.