The canonization process occurred during the period that currently comprises roughly the first quarter of the time that Christianity has existed, known as the Early Church.

The first specific list of canonical books was Marcion's, circa 140 (Marcion has since been condemned as heretical for various issues such as declaring the Christian God to be separate from the Jewish one). It contained some epistles and a modified version of the Gospel of Luke. Since then, there have been other canons proposed. Of specific lists, the extant one that most closely matches the standard 27-book New Testament is the Muratorian Fragment, with the canon mostly being determined through apostolic tradition.

One of the most important things to consider here is that when the bishops reach a consensus, they are considered to have been guided to their conclusions by the Holy Spirit. That is why ancient councils such as those at Nicaea and Constantinople were considered to be authoritative Nowadays, the most common occurrence of this principle is papal elections. However, last year's Pan-Orthodox Council was intended to be the same kind of thing for the Eastern Orthodox, but it seems to have had a lot of its own internal disputes. Additionally, the two Vatican councils hold this role for the Catholic Church (if you ever find traditionalist Catholics complain about "Vatican II," that is the second of these). The idea is that God would not let the Church fall into error (note that this is the proper-noun Church, not just any church, this refers to the One, Holy, Catholic, and Apostolic Church described in the creed that the councils of Nicaea and Constantinople came up with) and so would guide its leaders to a consensus on the truth (this same principle is the basis for the concept of papal infallibility determined at the First Vatican Council). This principle is why tradition is of very high importance in the Catholic and Orthodox churches (which make competing claims to be the Church in addition to being churches) in comparison to the importance of tradition in Protestant churches and denominations.

In summary, the canon was determined because over the first several centuries of Christianity's existence, the Church Fathers, the most influential religious leaders from that time, determined through tradition that 27 specific documents were authoritative.

There never was an official declaration saying that the canon was closed. Therefore, in theory, something like The Martyrdom of Polycarp could theoretically be added by developing a tradition of reading the stories of martyrs as an additional part of the Liturgy of the Word, in addition to the Old Testament, Psalm, New Testament, and Gospel readings that are the current components of the Liturgy of the Word. Don't count on it, though.

The most recent canonical declarations have been the result of the Reformation. Martin Luther attempted to drop four books (Hebrews, James, Jude, and Revelation) from the New Testament because of disputed authorship and because some of their stuff contradicted his interpretation of things in the Pauline Epistles. The most famous issue here is his objection to parts of James which contradict how he drew the concept of sola fida from Paul's epistle to the Romans. While this was not accepted, Luther's Bible translation into German (which is still in use, such as with the Amish) still puts them at the back of the New Testament. Luther additionally disputed a few portions of the Old Testament which are only found in Greek and not in Hebrew (again, the emphasis on tradition in the non-Protestant churches disagrees here, determining the Greek Septuagint to be authoritative because it was what the apostles used when they embarked on their Great Commission). These portions have, because of Luther's influence, been labeled "apocrypha" and either moved to the back or dropped entirely from most Protestant Old Testament canons.

In response to Luther's fiddling with the canon, the Catholic Church's Council of Trent (another one of those authoritative councils) included among its resolutions that the Bible MUST include the specific books that Luther had problems with in addition to the ones that he did not have a problem with. Notably, it did not say that it cannot include anything else, so the Council of Trent did not close the canon.

Keep in mind that this answer is specifically for Western Christianity. While Eastern Christianity often seems to hold tradition in an even higher regard than the Catholic Church does, there is more variance, particularly when you get to the non-Byzantine churches (Byzantine in this case being what would become the Eastern Orthodox). The Syriac and Ethiopian traditions in particular are well-known for having their own unique biblical canons.