None of the answers you're going to get to this question will be satisfying. Nor will any of them be correct.
I could tell you that "postmodernist" thought is generally characterized by a "rejection of objective Truth", and a strong suspicion towards "totalizing meta-narratives", but that probably wouldn't tell you very much. And it wouldn't be strictly accurate, either. I could tell you that it argues that many of the things which we take for granted are merely "social constructions", not merely figments of our imagination, but things that we as a society have indeed created for ourselves. I could say that it aims to expose and destroy oppressive systems of classification, particularly those that emphasize sharp divisions between groups—gender, race, culture, etc. But one of those things might give you the idea that it's somehow "pluralism", which is so completely wrong that I'm not even sure where to begin.
The reality is that "postmodernism" is not something that you can define, at least not as applied to philosophy. There were hardly any notable philosophers who accepted the label of "post-modernist", and many of those who are often considered to be anchors of the tradition actually strictly rejected the label.
Part of the problem, of course, is highlighted by Jon Ericson's answer: thinking that is typically associated with the "postmodern" camp seeks to reject things like labels and totalizing meta-narratives. They wouldn't be too keen on allying themselves with any particular camp, or labeling their unique brand of thought as "postmodernist". There is a notable tendency among these thinkers to resist the homogenization and categorization that such labeling implies.
About the only thing you can accurately say about "postmodernism" is that it's characterized by a rejection of modernism, the pseudo-scientific mentality of progressive objectivity established in the Enlightenment. Truly, one could argue that without the Enlightenment and the resulting movement that has become known as "modernism", there could never be such thing as "post-modernism". But whether that's actually a useful definition or merely a linguistic tautology is debatable.
And even if you accept that definition, a couple of problems still remain. First, it's extremely difficult to define something in terms of what it is not. Several notable "postmodernist" thinkers have taken up this very notion, albeit in quite different contexts. As merely one example, consider Jacques Lacan and his notion of the "lack". Second, several of those who are apparently "postmodernist" thinkers have actually accepted "modernist" notions to varying degrees. Jürgen Habermas in particular is very much a "modernist", and he doesn't even reject the notions of "universality" and "Truth" that seem to characterize the thought of most other "postmodernists". It's hard to say whether "postmodernity" actually seeks to replace modernity, to render it obsolete, or whether it merely allied with it, continuing and reinvigorating the modernist project.
Many, many, many "postmodernist" scholars vehemently disagree with one another. Somehow, this camp has grown to contain post-Marxists, feminists, deconstructionists, anarchists, post-Freudian psychoanalysts, and everything else outside and in between. Many people use it as a label of denigration, or at least a nicer way of saying "those people", the "different" ones, the ones "we" don't agree with. You see this in an unfortunate amount of academic literature, and in an even more unfortunate amount of colloquial rhetoric. See, for example, Paul Hartman's "What is 'Postmodernism'?" or the infamous Postmodernism Generator (refresh repeatedly for a real thrill!). In a truly postmodern sense, its accuracy lies in its appalling inaccuracy.
So yes, of course it has a fuzzy definition, and of course the Wikipedia article seems to characterize it in terms of what it is not. There's not much else that it can do. In fact, reading the Wikipedia article now, it strikes me as possibly one of the best attempts to clarify and define the movement that I've ever seen. And I've read a whole lot of so-called "postmodernist" literature.
Beyond that, the paragraph you mention concluding that postmodernism has influenced many other cultural fields is an important one. "Postmodernism" is not limited in scope to philosophy. In fact, when applied to other disciplines, an even more complicated web begins to emerge. From architecture to art to music, and dozens of other disciplines in between, "postmodernism" takes on very important meanings. Wikipedia links to a host of articles on "postmodern x", where x represents some particular artistic discipline. Again, what most of them have in common is the rejection of "modernism"
"Postmodernists" would tell you that you're really asking the wrong question. Rather than trying to put a label on their critical project, to attempt to unite and thereby destabilize it from without, you should join in on the project and become part of the movement.
If "postmodernism" implies anything at all, it implies problematization, the refusal to tacitly accept anything as objectively true, as objectively real, as objectively valuable. That doesn't, of course, mean that it embraces skepticism. That doesn't mean that it necessarily rejects all notions of objective truth. But it does mean that the goal is to problematize everything, to attempt to see everything from a different light, to root out problems at their source, to emphasize connections that haven't been previously observed or noted, to engage in relations with other cultures, values, practices, and systems of thought. It's a critical project, one in which nothing is taken as absolute, including the idea that nothing is an absolute.
