I understand the doctrine of inerrancy as the conviction that the Bible, when correctly interpreted, is without error in the original autographs. Note that there are two important caveats in this definition.
When Correctly Interpreted
First, the Bible is only without error when correctly interpreted. This is an important point which I believe is often misunderstood by both fundamentalists and skeptics alike. The concept of error cannot be divorced from the concept of interpretation. If one claims a text is in error, one is at the same time claiming to know something about what the correct reading would be, and is thus engaged in interpretation. Interpretation, furthermore, depends heavily on questions of genre, context, language, and culture.
For example, consider Psalm 75:3:
When the earth totters, with all its inhabitants, it is I who keep its pillars steady.
Now we know that the planet Earth is hurtling through space; it is not resting on pillars. Does Psalm 75 therefore contain an error? Only if we interpret this text as a scientific treatise on cosmology. If instead we correctly interpret this text as a piece of Hebrew poetry, filled with metaphor, we find no error.
Consider as another example the story of Jairus’ daughter. Recall that while going to the house, Jesus was approached by a sick woman who touched his garment and was healed. In Mark and Luke, we are told that after Jesus healed this woman, Jairus was informed that his daughter had died.
He said to her, "Daughter, your faith has made you well; go in peace, and be healed of your disease." While he was still speaking, some people came from the leader's house to say, "Your daughter is dead. Why trouble the teacher any further?" (Mark 5:34-35)
He said to her, "Daughter, your faith has made you well; go in peace." While he was still speaking, someone came from the leader's house to say, "Your daughter is dead; do not trouble the teacher any longer." (Luke 8:43-49)
In Matthew, however, we are told that Jairus was informed of his daughter’s death before Jesus encountered the woman.
While he was saying these things to them, suddenly a leader of the synagogue came in and knelt before him, saying, "My daughter has just died; but come and lay your hand on her, and she will live." And Jesus got up and followed him, with his disciples. Then suddenly a woman who had been suffering from hemorrhages for twelve years came up behind him and touched the fringe of his cloak. (9:18-21)
Even a casual read of the gospel narratives reveals dozens of such discrepancies. Are these errors? Again, it all depends on how you interpret the text. Just as Psalm 75 is in error if it is evaluated by the standards of a scientific textbook, the gospel narratives are in error if they are evaluated by the standards of modern journalism. However, while detailed precision is expected in our culture, ancient Greek did not even have quotation marks. Historians regularly abbreviated, rearranged, and paraphrased material. It was not considered improper.
Note that one can affirm inerrancy and still hold unorthodox beliefs. Consider the controversial issue of homosexual behavior, which Paul clearly condemns (Rom 1:21-27, 1 Cor. 6:9-10). Now someone might say that Paul was an ornery old coot who was just plain wrong. This person would, of course, be rejecting the doctrine of inerrancy. However, another person might affirm the doctrine of inerrancy and still maintain that homosexual behavior is acceptable. For example, they might say that Paul is talking about another kind of homosexuality, perhaps involving pedophilia or prostitution, and never envisioned the notion of a monogamous, loving homosexual relationship.Now I believe this view is wrong, but I cannot simply appeal to the doctrine of inerrancy to refute it. I have to do my homework and show that this interpretation of Paul is not plausible.
In the Original Autographs
The second caveat in my understanding of inerrancy is that it applies only to the original autographs. By an original autograph, I mean, for example, the actual, physical letter which Paul sent to the church in Rome. Of course, none of the original autographs of the Bible have survived the millennia. We do not, therefore, posses an inerrant text. What we possess are thousands upon thousands of copies and translations of these original manuscripts. No two copies are perfectly identical; they contain scribal errors and even intentional changes, such as attempts to harmonize the wording in the gospels.
However, through the science of textual criticism, we have been able to construct a good approximation of the original text. Textual criticism involves comparing variant readings and applying various criteria to discern which reading is original. As more and more manuscripts are discovered an analyzed, our text continues to improve.
The text of the New Testament is now well established. The modern text is based on many early manuscripts, including p66 (pictured above), which dates from the late second to early third century and contains almost the entire Gospel of John. Furthermore, the viable variants are relatively insignificant, and we know the options. In other words, we are not 100% sure what 1 John 1:4 originally said, but it either said, “These things we write, so that OUR joy may be made complete,” or, “These things we write, so that YOUR joy may be made complete.”
The Old Testament is a more complex matter. Due to the fact that the Old Testament is so much more ancient than the New Testament, there is a greater gap of time between the original composition of the text and the composition of the surviving copies. Furthermore, the text sometimes underwent significant development during this time. For example, the Hebrew text of Jeremiah used to produce the Septuagint (the Greek translation often quoted by the New Testament authors) was apparently about 15% shorter than the later Masoretic text (the “official” Hebrew text on which our English translations are based).
In conclusion, I think inerrancy is an important doctrine, but it is an "in-house" debate. You do not have to affirm inerrancy in order to become a Christian. You do not have to believe that the gospels are supernaturally inspired documents before you can consider Jesus of Nazareth.
If you are not yet a believer, I would encourage you to simply read the Biblical texts as you would any other ancient documents. Do not assume these texts are inerrant, and do not assume that they are the product of some vast, coordinated conspiracy. Read them with an open mind.