The basic enterprise of contemporary literary criticism is actually quite simple. It is based on the observation that with a sufficient amount of clever handwaving and artful verbiage, you can interpret any piece of writing as a statement about anything at all.
This is a degeneration of Derridas Deconstruction which could be viewed as an attack on the then dominant (& stagnant) school of Structuralism or a way past it. To use a mathematical analogy: mathematics (in one sense) is about axiomatic systems, but this does not mean that any axiomatic system is of equal value. Likewise not every interpretation of a piece of writing is of equal value. Judgements of taste must still be made.
The broader movement that goes under the label "postmodernism" generalizes this principle from writing to all forms of human activity, though you have to be careful about applying this label, since a standard postmodernist tactic for ducking criticism is to try to stir up metaphysical confusion by questioning the very idea of labels and categories.
Postmodernism is a questioning and reaction of Modernism; in the same way that Romanticism was a reaction to early Modernism. From some point in the future looking back it may be seen as part of Modernism. Its really too early to say (though of course one does).
"Deconstruction" is based on a specialization of the principle, in which a work is interpreted as a statement about itself, using a literary version of the same cheap trick that Kurt Gödel used to try to frighten mathematicians back in the thirties.
Deconstruction is roughly about inverting dominant modes of interpretation, in various modes, and its not a new technique: after all Marx inverted Hegel to present a critique of Capitalism. One could say that Deconstruction is both a literary & political tool.
Godels theorem, from a mathematical logic perspective is not a cheap trick, but certainly it has been used as a cheap trick by philosophical & mathematical hustlers. Paradox & antinomies have been used by serious philosophical thinkers, such as Hegel and Kant (in passing only) in the West; and by Nagarjuna and Daoism in the East.
Godels achievement, in context, is one part of the reinvigoration of formal logic since Frege, he introduced new techniques and questions into mathematical logic. However most popular expositions miss the importance of Paradox and tying it into the larger framework of Paradoxical thought in Philosophy - they settle for an exposition of Godels proof, whereas his main ideas are explicable in fairly simple terms - as they should be - and they do not give the larger & broader picture of Mathematical Logic: categorical Logic, intuitionist logic, inconsistent mathematics, paraconsistency and so on.
There is an incredible amount of verbiage about Godels Theorem, important though it is, which should be contemplated alongside the incredible amount of verbiage around Deconstruction, important though that is.
One of the elements of Badious Programme is to prune back this verbiage & metaphysical idiocy by making mathematics the site of ontology. But one should note that his book Being & Event references the Event of Derrida in the paper he presented at Columbia University which was to consolidate Structuralism but actually became a springboard for Deconstruction.
Although, Godels Theorem is presented usually as a death-knell of Mathematical Logicism, there has been found ways past it; certain parts of his programme has been completed. For example Gentzens proof of the consistency of PA, paraconsistent logic helps overcome contradictions in the rational architecture of mathematics by localising them.
There appears to be a general tendency towards Logical Pluralism which might be considered the outcome of the Logical Monism of Hilberts programme after a century of thought.
So far from Post-Modernism being inconsequential, one can see that the grand narrative of logical monism which may be seen as part of the modernist project has become Post-Modern by moving towards Logical Pluralism. Not the One but the Multiple.