As with the Old Testament, we just don't know who wrote most of the New Testament. Tradition has assigned the Gospels and most of the Epistles to certain authors, all of whom were important figures in Jesus' life or the early days of the faith. It was important for the early church to believe the authors wrote the works attributed to them, since their eminence lent the writings authority. But since we don't have the original signatures, none can be verified except through textual clues.

The first generation of Christians didn't see any need for a permanent written record of the sayings and stories of Jesus. Jesus' return and the restoration of the Kingdom of God on earth were imminent--why bother preserving stories if the world was about to end? Stories were simply passed along orally, primarily as a means of preaching and convincing outsiders. But as the first generation began to die off and hopes for the Second Coming dimmed, there was a need to preserve Jesus' words and deeds for posterity.

Quite a few collections of stories about Jesus circulated in the early church, among them The Gospel of Thomas, The Gospel of Mary, and the Secret Book of John. Some of these gave very different and in some cases conflicting accounts of the gospel and, most importantly, of Jesus' alleged resurrection. Some argued for the physical resurrection, with the mantle of leadership falling on those who had experienced it firsthand: the apostles. Others said the resurrection was a spiritual event that anyone could experience. Some thought this latter "heresy" would have led the church away from an organized entity into a situation where anyone could judge the truth for themselves. As Elaine Pagels points out in The Gnostic Gospel, this was no trivial matter. The decision on which interpretation was "correct" was central to the future of the church. 

We'll return to the question of how the "canonical" books of the New Testament were determined in the fifth and last installment of this answer. For now we'll just say that Iraneus, the bishop of Lyons in 180 AD, decided that the validity of any work had to be judged by whether it was "apostolic." That is, it should have been written by or for one of the twelve apostles. But, as Pagels goes on to say, regardless of whether the names given to the Gospels are those of the actual authors or merely reflect a claim to apostolic authority, "we know virtually nothing about the persons who wrote the Gospels."

Recent scholarship or, more correctly, recent rethinking of previous scholarship has brought an intriguing possibility to the table. Matthew, Mark and Luke are termed the Synoptic Gospels, so called because they generally agree on the details and timeline of Jesus' life, sometimes even using the same words to describe the same events. Because of this similarity, quite a few scholars posit that there was a previous collection of Jesus' sayings and works which all three gospel writers relied on when compiling their histories. This collection, as yet just a theoretical construct, has been given the name "Q" (short for Quelle, German for "source").

It's a tempting idea. Mark is regarded as the earliest gospel and hence closest to Q. Of the 661 verses in Mark, only 24 aren't quoted in either Matthew or Luke. Matthew and Luke occasionally disagree with Mark regarding Jesus' words or the order of events, but they never both disagree on the same point. 

Burton Mack in The Lost Gospel: The Book of Q and Christian Origins offers another conjecture. It's possible Q was the work not of a single person, but rather of a community trying to give written form and substance to what it believed. If that's the case, the question of authorship in the usual sense evaporates. But rather than have this discussion come to an abrupt end, we'll work on the assumption that the authors were individuals, not a committee.

Mark, not an apostle himself, was an associate of the apostle Paul for a short time, but the gospel bearing his name is (to some minds) based on the preaching of Peter. It's generally assumed to have been the first gospel written, coming in right before Matthew at about 65 AD.

The author of Matthew is traditionally held to be the tax collector mentioned in Matthew 9:9, sometimes referred to as Levi. However, Matthew borrows heavily from the Gospel of Mark. It's hard to believe someone who was in close contact with Jesus would have had to rely on secondary sources. Since this gospel has the most quotations from the Old Testament, sometimes going to ridiculous lengths to try to show that Jesus was the fulfillment of Old Testament prophecy, it's assumed that Matthew was written for a Jewish audience. There is suspicion that it might have been originally written in Hebrew, although only Greek texts have ever been found. Scholars differ on the composition date, but most agree on roughly 65 - 70 AD with a few placing at as late as 100 - 134 AD.

The Gospel of Luke and the Book of Acts are assumed to have been written by the same person, since they are addressed to the same individual, a Roman named Theophilus. The author was a doctor, Paul notes in Colossians 4:14. If Mark represents the teachings of Peter about Jesus, Luke most likely represents the teachings of Paul. Luke claims to have researched his material, but his dating, especially in the early chapters regarding Jesus' birth, is inconsistent with other sources.

The book of Acts can be seen as a sequel to the gospel of Luke, starting where the previous book ends. But where in the earlier work Luke needed to research the story, in Acts he is a character in it. He was a companion of Paul on his missionary journeys and was present during his imprisonment. In this sense, Luke had more first-hand experience of Paul than he had of Jesus. Both books were probably written after Matthew and Mark, probably around 65-70 AD but before the destruction of Jerusalem in 70 AD.

The Gospel of John differs markedly from the other three books both in tone and in some historical details. John does not follow the timeline in the other three and adds quite a few stories and details not found in them. For this reason, it's thought that John's gospel was not a child of Q, but a completely original work either by someone who knew Jesus directly or by one of his associates. The three letters of John found near the end of the New Testament are generally assumed to have been written by this same individual.

The identity of John has remained a mystery, although tradition has it that he is "the disciple that Jesus loved" mentioned in John 13:23. But here is a curious thing. In the entire gospel, John never mentions his own name (although he does mention other gospel writers). His purpose is to exalt the deity of Jesus. It seems out of character for him to pat himself on the back in that one verse, if in fact he was John the apostle.

William Barclay gives us an elegant answer. He states outright that even if John was not the direct author of the book, it was at least written under his authority. The book likely dates from about 100 AD, the last of the books to be written. If this dating is accurate, John would have been very old. Barclay posits that it was probably a group writing remembrances from John's fading memories, and it was they who described John as the disciple Jesus loved..

The letters to the Romans, the Corinthians, the Galatians, the Ephesians, the Philippians, the Colossians, the Thessalonians, Timothy, Titus and Philemon are widely assumed to have come from the hand of the apostle Paul and are called the Pauline epistles for that reason.

E. P. Sanders says it's fairly clear Paul was unaware of the four Gospels, and the authors of the Gospels didn't know of Paul's letters.

A few small stylistic variations in Colossians and Ephesians make some scholars suspect Paul didn't write them, but the evidence is sparse and unconvincing. The letters to Timothy and Titus are suspect as well, and some critics feel they were later edits of some of Paul's more personal correspondence to individual church leaders, or pastors.  Hence, they are often referred to as the Pastoral epistles. 

The author of the letter to the Hebrews is completely unknown. Stylistic or literary criticism has failed to match it with any known author, although it is usually included among the letters of Paul. Some names that have been bandied about are Barnabas (an associate of Paul), Apollos, or even a dual authorship of Aquilla and Priscilla, two Christians who ran a church out of their house in Rome. Early tradition knew that it was anonymous, but since it was such a popular work among the early Christians, it was included among the letters of Paul in order to insure its apostolicity and thus its place in the Bible.

The letter of James isn't anonymous, but it's not known who exactly James was. Five people named James are mentioned in the New Testament, one of whom was the brother of Jesus. It's this person whom tradition has accepted as the author, although the evidence is sketchy.

It's always been assumed the first and second letters of Peter were in fact written by Saint Peter. No real objection to that belief has been raised until rather recently, largely because few early church fathers quoted it as they did other canonically accepted books.

The Revelation is often called the Revelation of Saint John. Tradition says this is the same as the author of the fourth gospel, but that seems implausible. The style of the Greek is different, and while the gospel author avoids mentioning his own name in order to focus attention on Jesus, the author of Revelation mentions his own name repeatedly. He doesn't call himself an apostle, as would be his right, but merely a prophet. Exactly who the author was is open to conjecture. There is no real consensus, except that he was apparently a Jewish writer, writing in Greek to the Jewish believers after the destruction of Jerusalem in 70 AD. Most critics put the date at about 95 - 100 AD.

RESOURCES:

The Literary Guide to the Bible, edited by Robert Alter and Frank Kermode, Belknap Press, 1987
The Gospel of John, by William Barclay, Westminster John Knox Press, 1975
The Unauthorized Version: Truth and Fiction in the Bible, by Robin Lane Fox, Knopf, 1992
The Lost Gospel: The Book of Q and Christian Origins, by Burton Mack, HarperSanFrancisco, 1993
Testament, by David Morell, Warner Books, 1993
The Synoptic Gospels, by Keith F. Nickle, John Knox Press, 1980
The Historical Figure of Jesus, by E.P. Sanders, Penguin Books, 1993
The Catholic Encyclopedia