Form criticism begins with the assumption that human beings are not generally in the habit of memorizing immense blocks of text. Rather, we prefer to tell short, memorable stories with predictable structures and amusing punchlines; in this, our favourite anecdotes ("what I did on my summer vacation") do not differ much structurally from jokes, fables, or proverbs.

During the first half of the twentieth century, a number of European scholars tried to imagine how such an understanding of narrative might inform the study of the Gospels. Rudolf Bultmann was probably the most important of these scholars, though Martin Dibelius and Karl Ludwig Schmidt also contributed a great deal to the development of form-critical methods.

Form critics believe that the Gospels are loose collections of short, easily-memorized narrative snippets, compiled and rearranged in a way that served each evangelist's agenda. In the Bible, each of these snippets is called a pericope. Bultmann believed that most of the pericopes found in the New Testament could be categorized as belonging to one of a limited number of forms (hence the name "form criticism"). The most important of these forms are the following:

1. Pronouncement Stories, also known as paradigms or apothegms.
   These are brief stories which culminate in an authoritative saying. For example, when the Pharisees complain about the fact that Jesus is eating with tax collectors and sinners in Mark 2:15-16, Jesus silences them by announcing, "It is not the healthy who need a doctor, but the sick." Jesus' point is made pithily and memorably; if he is in conflict with some sort of opponent, he is always made to look wittier and more composed than they are. The pronouncement story is related to the classical Greek rhetorical technique known as the chreia.
2. Miracle Stories.
   These are self-contained, highly descriptive stories about a healing (e.g. Mark 2:3-5), an exorcism (e.g. Mark 5:1-13), or some other type of miracle. Miracles that are difficult to categorize -- such as the feeding of the five thousand, the only miracle that appears in all four gospels -- are often given the catch-all name "nature miracle." There are dozens of miracle stories in the New Testament; often they have some point to make about Christology, the kingdom of God, the meaning of true discipleship, or somesuch.
3. Logia, also known as dominical sayings. The word logion simply means "saying" (logia is plural). Bultmann classified Jesus' sayings into a number of specific types, including:
   • wisdom saying or proverb, such as "Foxes have holes, and birds of the air have nests, but the Son of Man has nowhere to lay his head" (Matthew 8:20).
   • prophetic or apocalyptic utterances, such as "Nation will rise against nation, and kingdom against kingdom..." (Mark 13:7-8).
   • "I" sayings, sometimes known by the Greek phrase ego eimi "I am". These are statements in which Jesus makes a memorable observation about his own nature, especially popular in John's gospel. e.g., "I am the bread of life" (John 6:48); "I am the door" (John 10:9).
   • legal sayings and church rules
     These are not always "legal" in the sense that we might think of the word in modern English. Rather, they are quotations from the Torah or creative interpretations of it. For instance, "You have heard it said to the men of old, 'You shall not kill; and whoever kills shall be liable to judgement.' But I say to you: everyone who is angry with his brother shall be liable to judgement" (Matthew 5:21-22).
4. Parables, which is to say short metaphorical or allegorical narratives with startling and sometimes counter-intuitive conclusions.
   Sometimes these are further subdivided into "similitudes," "example stories", and "parables" proper, but I personally can't be bothered with worrying about details like whether or not the story's in the past tense. Anyway, Jesus is famous for his parables, even among people who are not Christian. "The kingdom of God is like a mustard seed..." (Mark 4:30-32 and parallels).
5. Myths and Legends.
   These are stories with a supernatural element, such as can be found in the baptism or the Transfiguration of Jesus. Despite the name, it is not necessary to assume these stories are false; however, Bultmann often did.
6. Speeches.
   Quite a few characters deliver speeches in the New Testament, not only in the Gospels but also in Acts. However, since this entire categorization system arose out of the desire to find the Historical Jesus, it is Jesus' speeches that usually get the closest attention. Speeches in this sense tend to be significantly longer than both logia and parables, though Bultmann believed that they were originally made up of shorter bits that once circulated independently before being crammed together by the evangelists. An example of a speech would be the lengthy "farewell discourse" that fills John 14-17.

The driving assumption behind form criticism is that these small narrative units were rearranged and rewritten by the evangelists in ways that often obscured the original historical saying or event that lay behind them. Bultmann, despite being a committed Christian and serious theologian, was notoriously skeptical regarding the reliability of the gospels. He applied what we might call a "guilty until proven innocent" hermeneutic to most of the texts he studied. He (and the form critics who followed him) tended to see the evangelists as thoughtlessly cutting and pasting the traditions they inherited in artificial ways, cheerfully adding misleading "interpretations" of their own to Jesus' words. For instance, a form critic would argue that the parable of the sower in Mark 4:1-9 was originally delivered without any explanation at all; it was the evangelist, not Jesus, who added the secondary interpretation of the story's symbolism in 4:13-20. Needless to say, this angered a lot of traditional Christians who maintained the inerrancy of the Bible.

Today, form criticism is not as fashionable as it once was, but its insights have served as the building blocks for newer, more sophisticated theories. After the Second World War, form criticism was replaced with other interpretive techniques, particularly redaction criticism. However, even though some of Bultmann's assumptions have been rejected in recent years, is nevertheless impossible to engage in serious study of the New Testament without understanding his analyses.

Form criticism is sometimes known by its German name, formgeschichte, even among English speakers, since the technique was designed and promoted primarily by scholars in Germany.

Further Reading:
A good introduction can be found at http://www.tf.uio.no/lo/lck/blomberg0192.html.
However, nothing beats a close reading of Bultmann himself. Try his History of the Synoptic Tradition (1921).