While the Old Testament covers a period of well over a thousand years, the New Testament covers less than a century. It consists mainly of gospels (4) and letters (20 or 21) written by the first generations of Christians following Christ's death and resurrection as they struggled with all the problems of bringing a new faith and church into being.
The earliest writings are some of Paul's letters, with other letters and the gospels written later. Letters were a familiar form, but the gospels are quite new. The word 'gospel' translates the Greek evangelion, or good news, and it may be that the gospels began as oral accounts of the good news of Jesus Christ, until Mark first wrote down the story of that good news. More than four were written, but only the four we are familiar with made it into the canon of the New Testament.
We call it the 'New' Testament, because the people who wrote it were formed by the 'Old' Testament. There is a clear continuity between the story of God's people Israel, and the story of his people who became the church, with the old covenant, or the Law, transformed and fulfilled in the new covenant made through the life and death and resurrection of Jesus Christ.
As with the Old Testament, the rich history of the writings which make up the New Testament mean that there are discrepancies between different accounts. This is not really surprising, given the number of people who must have been involved, and the time span. Imagine a dozen or more people writing letters and gospels and other literary forms over a period of some 70 years, and you perhaps get some idea what it must have been like! The New Testament was written in Greek, but Jesus and his disciples would have spoken Aramaic, so add to that issues to do with language.
Matthew and Luke knew and used Mark's gospel, whereas John's is quite different in style and content. Many of the letters were written by Paul during his missionary journeys and when he was in prison in Rome. It is probable that Paul was executed in about 67AD, and that Mark's gospel was written around 70AD, to give some idea of time-scale.
It may be that not all the letters attributed to Paul were actually written by him. The letters to Timothy and Titus are generally believed to have been written later, and it is possible that Ephesians, Colossians and Second Thessalonians were also not written by Paul himself. There are also a number of letters which are not part of the New Testament which bear Paul's name, but which scholars believe to have been written by others. Hebrews does not bear Paul's name and it is highly unlikely that he wrote it.
In addition to the gospels and letters, we have the Acts of the Apostles, a second volume by Luke describing the period of formation of the early church, the Revelation, a work of apocalyptic, and Hebrews. Hebrew is conventionally described as a letter, but it actually reads much more like a sermon than a letter. Apocalyptic was well known in the centuries immediately prior to the birth of Christ, as we see in the book of Daniel in the Old Testament and in some of the books in the Apocrypha (a collection of books later than the establishment of the canon of the Old Testament, with some overlapping the New Testament in time). There are also short passages of apocalyptic in Mark 13, Matthew 24, Luke 21 and 1 Thessalonians 4.16-17. Apocalyptic is written with a sense of acute crisis - the end is upon us, we are in God's time now.