The first of this series was Faith, Evidence, and Biblical Scholarship. Here is the second New Testament post.
When we are talking about copies of the New Testament, a manuscript is a handwritten copy, most of which certainly date to the medieval period before the invention of the printing press in the late 1400s. As you would expect, we have fewer truly old manuscripts than we do later ones because of the wear and tear of time. We have none of the original manuscripts of any biblical writing and the vast majority date later than AD1000.
Some find this situation unnerving, but we are in a much better position when it comes to the New Testament than we are other ancient writings like those of Homer or Plato. We have a fragment of John that dates to about 30 years after it was written--almost unheard of. We have a collection of most of Paul's writings (minus the Pastorals) that dates to around AD200 (p46, one of the Chester Beatty papyri)--also quite amazing when it comes to such things.
The best introduction to the science of the textual criticism of the New Testament, the branch of biblical studies that studies manuscripts and tries to identify what the books of the Bible originally said, is Bruce Metzger's The Text of the New Testament.  Metzger is also responsible for the very helpful, A Textual Commentary on the Greek New Testament.  This resource goes through the most significant variations among the thousands of manuscripts we have and explains why the editors of the standard Greek New Testament picked the particular options they did.
Another good introduction is Kurt Aland and Barbara Aland's, The Text of the New Testament.  Kurt Aland was instrumental in putting together the standard text of the Greek New Testament as it now stands. There are two primary versions, although both now have the same text. The "Aland" version is typically a red Bible with a "textual apparatus" at the bottom to tell the most significant variations from the text the editors have chosen.  The "Nestle" version is typically a blue Bible and has been around longer.  It now has the same text as the "Aland" version, but a much more extensive textual apparatus at the bottom.
The "external evidence" we have, the evidence from various manuscripts and writings, falls into about five different types. First are papyri, manuscripts written on scrolls made from papyrus strips. They are typically the oldest manuscripts and are written in all capital Greek letters without any breaks between words or punctuation. The punctuation we now find in our Bibles is thus entirely a matter of the judgment of interpreters. It is thus possible to argue from time to time that a statement should really be a question or that something is a quote from somewhere else rather than an author's claim or that a period should be put somewhere so that a new thought begins.
"Uncial" manuscripts tend to date from about the 300s to the 800s and are on animal hide or "vellum." Rather than scrolls, uncials are in book form (i.e., in codex form), making it possible to have the entire New Testament on them. Two of the most important uncial manuscripts date from the early 300s, Codex Sinaiticus (symbolized by the Hebrew letter א) and Codex Vaticanus (symbolized by B). Although the standard Greek text currently in use does not read exactly like these two, most textual scholars believe these two manuscripts are examples of the best tradition of manuscripts, usually called the Alexandrian tradition.
"Minuscule" manuscripts are written in lower case Greek cursive and are medieval copies, largely dating from a handful in the 900s to around 1500 when copies began to be made with the printing press. We have well over 5000 of these. By the Middle Ages, the text of the New Testament had become relatively standardized, so most of these fall into what scholars often call the Byzantine tradition. When people speak of the Majority Text, they are largely referring to this textual tradition. Most surviving manuscripts are medieval and most read very similarly.
We have two other very important witnesses to the text of the New Testament. One are the early church fathers, the quotations that various significant figures in Christianity made of the biblical text. These fathers are as old as the oldest manuscripts and thus give an important witness to the text of the New Testament, although we have to be careful because all we have of their writings are also copies of copies. We thus have to do textual criticism on manuscripts of them just as on manuscripts of the New Testament itself. It is understandable if some of their copyists may at times have harmonized their quotes with the text of the Bible as these copyists knew it in their own time.
A good example of the significance of quotations in the fathers is the ending of Mark, Mark 16:9-20. The manuscript evidence for this ending being original is not very good. The oldest copies we have of Mark do not have it. Several early Christians do not seem to know about it and two very important Christians from the 300s and 400s say that few manuscripts of Mark they knows have it. But at the same time, two very important witnesses of the late second century, Irenaeus and Tatian, seem to know of the ending. The strongest argument from the "external evidence" of manuscripts, therefore, comes from early church fathers.
A final major witness to the text of the New Testament are translations of it into other languages like Latin, Syriac, Coptic, Ethiopic, Armenian, and so forth. Again, some of these translations, like the Old Latin, can be as old as any Greek manuscript. While one may not always be able to tell the exact wording of the Greek original, they can help a person choose between known Greek options.
The science of textual criticism primarily developed over the last two hundred years. The 1800s in particular saw significant findings of old manuscripts that called into question the originality of the "received Greek text," what had come to be called the textus receptus. The origins of this particular edition of the Greek New Testament, which stands as the basis for the King James or Authorized Version, is quite interesting.
With the invention of the printing press, the humanist, Renaissance, Roman Catholic scholar Erasmus embarked on a race to be the first to put the Greek New Testament in movable type. Using a half dozen or so manuscripts, all of which were medieval, he quickly put together an edition of the Greek New Testament that understandably corresponded mostly to the Byzantine tradition. For his first edition, he did not have any copies of the last part of Revelation, so he actually created a Greek text by translating back from the Latin. The result was some readings that ironically appeared in not a single Greek manuscript!
By the 1551 edition of Stephanus, the Erasmus text had reached somewhat of a standard form. For the first time, chapters were divided into verses. These verse divisions remain the basis for the current numbering in the New Testament.  In a few instances, later findings of earlier manuscripts have suggested that a verse of Stephanus' text was not originally there. The result is a "missing verse" like Acts 8:37. But no one has taken this verse out. Rather, the earliest manuscripts suggest instead that the verse had been added in centuries after Acts was written.
In a few instances, the textus receptus has retained some of the peculiarities of Erasmus' venture. For example, the reading in Revelation 22:14 in the King James Version, "they that do his commandments" is a leftover from Erasmus initially translating back from Latin into Greek because he did not have any Greek manuscripts here. There are some Greek manuscripts that read this way, but we wonder if he would have gone with the more likely, "those who wash their robes" if he had initially had any Greek manuscripts of this verse at the time.
Perhaps the most famous Erasmus story has to do with 1 John 5:7. The full verse appears in only eight of the over 5000 known Greek manuscripts. It appears in the main text of no known Greek manuscript prior to Erasmus. If original, this wording of the verse would be the clearest statement of the Trinity in all of Scripture. Accordingly, Erasmus received some pressure to put it in a later edition of his text. He remarked that he would if a Greek manuscript could be shown to have it. His journal later suggested he believed the one he then received had been created for that very purpose. Since the verse has such incredibly weak support and in fact was unknown at the time of the great Trinitarian controversies of the 300s and 400s, it seems incomprehensible that they be original.
In the centuries after Erasmus, there was an increasing sense that his Greek text was not original at various places. J. J. Griesbach (1745-1812), for example, suggested that in general, the shorter of two variations (the lectio brevior) is more likely to be original. It is not always the case, but more often than not. He also recognized the very important principle that the more difficult reading (the lectio difficilior) is more likely to be original. The reason is that copyists were more likely to "fix" problems than to cause them when copying.
Hiding here is probably the most important principle of textual criticism of all, at least when it come to internal evidence, evidence based on the way the text reads rather than on what manuscripts have a certain reading. This is the rule that says the most likely original reading is the one that would best explain how the other variations came about. Because copyists were more inclined to add material to clarify things or to smooth out difficulties, the shorter readings and the more difficult readings most of the time are the more likely readings, and the older manuscripts tend to have these readings.
One of the greatest discoverers of manuscripts was Constantin von Tischendorf (1815-74), the "Indiana Jones" of manuscripts. His biggest discovery was Codex Sinaiticus, the oldest complete Greek manuscript of the entire Bible known at present. It dates from the early 300s. Tischendorf came across it while staying at St. Catherine's monastery on the Mt. Sinai peninsula, where it was in a bin of material destined to be burned for heat. He strongly suggested it should not be burned and, by way of politics, eventually managed to get the manuscript out of Egypt and to him in Germany.
Some of the most groundbreaking discoveries both of manuscripts and of method were thus in play by the late 1800s. But it was undoubtedly the work of Brooke Foss Westcott (1825-1901) and Fenton John Anthony Hort (1828-92) that finally established the current trajectory of New Testament textual criticism. In 1881, they published a critical edition of the Greek New Testament that incorporated the consensus of textual scholars as it had developed over time. Their second volume laid out the method currently in use of weighing variations against each other on the basis of both internal and external evidence. In terms of external evidence, they divided existing manuscripts into various traditions, the most important of which were the Alexandrian that they preferred (e.g., Sinaiticus and Vaticanus); a Western textual tradition whose principal representative was Codex Bezae (D) and some Old Latin manuscripts; and the Byzantine tradition that most surviving manuscripts exemplify.
The science of textual criticism has not stood still since the days of Westcott and Hort. It is generally agreed that they relied too heavily on the two manuscripts, Sinaiticus and Vaticanus. Nevertheless, older papyri have been discovered since them and these finds generally support the conclusions of Westcott and Hort. Certainly we have also seen a few scholars who have applied their intellect to finding ways to reinterpret the evidence. One such attempt focuses on the "Majority Text" of the Greek New Testament, considering the reading that most manuscripts have as more likely original than the reading favored by "weighing" the manuscripts by age, tradition, and internal features. However, by far the vast majority of textual scholars accept the dictum of Westcott and Hort that "manuscripts must be weighed, not counted" and would thus consider the text used in most modern translations of the Bible to be by far the most likely original wording of the New Testament books.
 (Oxford: Oxford University, 1964). The fourth edition has been revised by Bart Ehrman (2005).
 (London: United Bible Societies, 1971).
 Translated by Erroll F. Rhodes (Grand Rapids: Eerdmans, 1987).
 Put out by the United Bible Societies, now in its fourth, revised edition (2006).
 Put out by the American Bible Society, now in its 27th edition (2004).
 The chapter divisions had only been added in the early 1200s by a man named Stephen Langton.