How were the books of the New Testament chosen? I'm interested in finding out:


what the process of choosing the books of the New Testament was, 
when did it occur,
who did the choosing, and
what are the supporting evidences
The canon developed gradually over the course of more than 300 years. In many cases, when decisions were made, they were simply to acknowledge what was already being read in the churches.

The process started early. Already in 2 Peter 3:16, there is a reference to the letters of Paul:


  There are some things in them hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other scriptures.


So at least some of the churches were already circulating Paul's letters and reading them as scripture.

The Gospels were written after Paul's letters, and the Apostolic Fathers quoted most often from Matthew, but also sometimes from Mark and Luke, and eventually from John.

By the late 2nd century, Irenaeus (Against Heresies 3.11.8) was claiming that the canon must contain exactly four gospels:


  It is not possible that the Gospels can be either more or fewer in number than they are. For, since there are four zones of the world in which we live, and four principal winds, while the Church is scattered throughout all the world, and the “pillar and ground” of the Church is the Gospel and the spirit of life; it is fitting that she should have four pillars, breathing out immortality on every side, and vivifying men afresh.


This was to counter fringe groups that were producing their own gospels, as well as Marcion, who argued for just one gospel (Luke).

We don't know who compiled the list known as the Muratorian Fragment, but it also dates from the second century and contains this canon:


Four gospels (the existing fragment begins by naming Luke as the third and John as the fourth)
The Book of Acts
Thirteen letters of Paul (and then names two letters attributed to Paul but not accepted as genuine)
Jude
Two letters from John
Revelations "of John and Peter"
"the Wisdom written by friends of Solomon in his honour"


Hebrews, James, 2 Peter, and 3 John were not yet accepted.

The fragment also recommends the Shepherd of Hermas as being worth reading but not qualifying for the canon because it was written "quite lately in our time".

By the early fourth century, the church historian Eusebius (Church History 3.25) sorts the known early writings into cateogories.

The first category is:


Four Gospels plus Acts


Next in importance are:


The letters of Paul
One letter each from John and Peter


These are the writings that are universally accepted by the church. Eusebius adds that some believe Revelation should be included in this set.

The next category includes books that are disputed by some:


James
Jude
2 and 3 John
2 Peter


These would all eventually be added to the canon.

The next category includes books that rejected from the canon but worth reading:


Acts of Paul
The Shepherd
Apocalypse of Peter
Letter from Barnabas
The Teaching of the Apostles


He mentions that some would place Revelation in this group, and others would include the Gospel of the Hebrews.

And finally, Eusebius mentions other writings that are considered heretical, which he says should be "cast aside as absurd and impious." These include the gospels of Peter and Thomas, among others.

The first known list of canon that matches today's New Testament is found in the Easter Letter of Athanasius for the year 367:


  Again it is not tedious to speak of the [books] of the New Testament. These are, the four Gospels, according to Matthew, Mark, Luke, and John. Afterwards, the Acts of the Apostles and Epistles (called Catholic), seven, viz. of James, one; of Peter, two; of John, three; after these, one of Jude. In addition, there are fourteen Epistles of Paul, written in this order. The first, to the Romans; then two to the Corinthians; after these, to the Galatians; next, to the Ephesians; then to the Philippians; then to the Colossians; after these, two to the Thessalonians, and that to the Hebrews; and again, two to Timothy; one to Titus; and lastly, that to Philemon. And besides, the Revelation of John.


Athanasius then adds that other books&mdash;the Wisdom of Solomon, the Shepherd, the Teaching of the Apostles, et al.&mdash;should be read by new converts. So even as the canon was being defined, there was recognition that among the excluded books were some that were part of the church teaching, and some that were not.

Following Athanasius, we see the beginning of a general agreement throughout the church. The Synod of Hippo in 393 published a list of New Testament books identical to Athanasius' list, but also included an Old Testament canon. The Council of Carthage of 397 published a similar list, but is notable for separating Hebrews from the letters of Paul. The Latin Vulgate translation, commissioned by Pope Damasus in 382 and completed by Jerome in 405, contains the same set of books listed in these. The Decretum Gelasianum, widely thought to be associated with the same Pope Damasus, lists the same 27 books but makes a distinction among the letters of John ("of the Apostle John, one letter, of the other John the Elder, two letters").

The Council of Carthage of 419 lists the same books, but does not make a distinction between two writers named John, and does not separate Hebrews from the letters of Paul.

In the East, there would still be disagreement for more than a century. The Council of Laodicea of 364 accepted all of the current New Testament books except Revelation. The Apostolic Constitutions, compiled around 380, has a Bible canon including this same list, but adding three writings attributed to Clement of Rome.

The first version of the Peshitta, the Syriac translation of the Bible, omitted the books of 2 Peter, 2-3 John, Jude, and Revelation; these books would not be translated into Syriac until the 6th century.

The Quinisext Council or Council in Trullo was held in Constantinople in 692 to reaffirm the rulings of the Fifth and Sixth Ecumencal Councils; while this council did not publish a formal Bible canon, it did reaffirm the canons previously published, including those of Athanasius and others with the Book of Revelation. This set the stage for the 7th Ecumenical Council (Second Council of Nicea) in 787, which finally established what we now know as the 27-book New Testament canon for the entire church.