Gary Aylesworth is right in pointing out that defining postmodernism is a very difficult, (maybe nearly impossible) task. Because it's such a huge and somewhat ambiguous topic with each person having their own take on it, it's difficult to talk about what it actually is (that in and of itself captures part of the mood of postmodernism).
A helpful way to understand postmodernism is in this way: postmodernism is that which stands over and against modernism. The term "postmodernism" gets the name because it means after modernism. So, postmodernism is a certain philosophical practice/method/lens which stands in contrast to modernism and also critiques modernism.
So, then we have to ask the question, "what is modernity?" In the history of philosophy, modernism is that period which contains thinkers like David Hume, Descartes, Immanuel Kant, John Locke. Modernity has been characterized as having a very analytical and rational (I mean using reason and thinking, I don't mean rational as opposed to empirical) approach to reality. Reality is out there, and all we need to do is to use our brains and study it, and then we can know it as it is, as an unbiased and unaffected observer of reality. If we use the methods of science, logic, and rationality, we can have this objective matter-of-fact knowledge. Modernity places a high emphasis on objectivity, proof, and propositional truth (that is to say that truth is expressed through propositions).
Postmodernism stands against all that. Postmodernism says, look, we are all part of this thing that we are trying to study and understand, we are in it and of it, so there's no such thing as being able to stand back and simply objectively look at reality as though one can have a God's eye view. We all bring our stories and presuppositions to whatever it is we are looking at. Thus, there can be no unbiased and objective point of view. Truth can't be just be propositional because we have to ask the question, "whose truth?." Truth must be beyond propositions. Additionally, the sources of knowledge are not limited to the realm of logic, rationality, science, etc. Postmodernism values narrative, personal experience, art, and the ineffable. So, for example you can read an argument with the thesis, "Seemingly insignificant persons can do significant things," or you can watch a movie about two hobbits on a quest to destroy evil. Finally, part of the postmodern mission is to reconsider and critique how things are done, seen, and interpreted. Postmodernism says, let's hear what every individual person thinks about x because they each have an interesting vantage point. One can see this for example in the feminist movement which often argues that things are and have been seen in a way that is very patriarchal. The feminist wants to critique that patriarchy and offer something new in its place. Additionally, a lot of the emergent/emerging church movement owes itself to postmodernism. It's a rethinking of how church has always been done, and it's trying to do something new. All of this is tied up in postmodernism.
So is Christianity compatible with postmodernism? First, to be fair, there are smart people on both sides of this issue. James K. A. Smith is probably the most popular Christian philosopher who embraces postmodernism. But there are also folks like Doug Geivett and Doug Groothius who would strongly reject postmodernism. I do think we have to take into consideration Scripture as a narrative. After all, God doesn't come down and hand us a giant, systematic theology book full of syllogisms, arguments, proofs, etc. Rather, in the Bible, we get a story. The story of God and God's people, with Jesus at the center. God also reveals Himself ultimately in a Person: Jesus. And Jesus himself went around telling stories. (On the other hand, the Bible is given to us through words. Indeed, God thought it best to give His people written words, and those written words have been preserved and lasted us all these thousands of years. Additionally, while the whole of the Bible can be seen as a story, there is a fair amount of "systematic" type theological language that we find in Paul's epistles for example.)
I also think postmodernism's critique of modernism's absolute scientific and objective approach is fair. There's no such thing as viewing something with perfect objectivity. Each person is a human being who does not exist in a vacuum. We all interpret through our stories and with certain presuppositions. If in the modern period, the goal of Bible scholars and theologians was simply to stand afar and critique the text, postmodernism asks how does the text critique us as the readers?
Finally, I think postmodernism's emphasis on knowledge through means other than cognitive reason is good. Postmodernism highlights that we can know things even if we cannot explain it, write it down, or completely logically understand it.
That being said, if postmodernism is reduced to the complete rejection of objective truth, and everything becomes about a matter of mere interpretation/perspective, then I think it is no longer compatible with Christianity. And I think this is what many Christian thinkers are afraid of. For example, the famous postmodern response goes, "You only believe that because...(fill in the blank: you're a man, you were born in the West, you grew up in a Christian home, etc.)". And the counter-response should be, "You only believe that because you're a postmodernist!" It's a very sneaky way of not having to actually deal with the argument itself. And of course, it commits the genetic fallacy which states that just because you can show how a belief originated, that does nothing to say whether or not the belief is true. Moreover, certainly there are some truths which are objective: "It is wrong to torture small children for fun." "It is good to love one's neighbor." "I am sinner." "Jesus is Lord." Surely these must be objectively true statements. And without certain objective truths and values, we just become lost in a sea of socio-cultural relativism. Why judge anything at all as good, bad, better, worse, true, or false? Finally, some postmodernists view language as just word games, and there is really no objective meaning in the words. Meaning is whatever we make of it. Though it's true that meaning exists where there are persons, to strip objective, true meaning altogether I think becomes incompatible with Christianity. Because then we face issues like people meaning whatever they want the Bible to mean, and that's problematic.
With that said, I don't think postmodernism has to be reduced to these negative qualities. I think we can glean from the positive qualities it has to offer. There is a lot that postmodernism has done that is helpful and important. So, to dismiss postmodernism altogether as anti-Christian, I don't think would be very fair to it.
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