What is "deconstruction"? As French philosophy professor Jacque Derrida (1930-2004) used the term, it means: analyzing a written work, such as an essay or a book, to uncover the always corrupt assumptions that underlie all Western philosophy.
Derrida was a founding member of the Post-Structuralist movement. That was the movement that rejected the earlier 20th Century Structuralist movement. Structuralism was an attempt to apply to specialized sciences -- such as anthropology -- a foundationalist (Cartesian) epistemology. Structuralists believe we can organize all of our knowledge into one syllogistically consistent structure whose foundation is fundamental concepts explicitly defined. For the Structuralists, example fundamental concepts (presented as antitheses) are: truth versus falsity, and presence versus absence.
In Objectivist terms, Structuralism is an instance of extreme rationalism. (See The Ayn Rand Lexicon for a brief definition of "Rationalism.")
Derrida's purpose was to show that Structuralism's antitheses (he calls them "binary oppositions") are not dichotomous but intertwined. He analyzed texts -- e.g., written by Plato and Rousseau -- to show that the authors were actually undermining the very concepts they supposedly were applying.
What was Derrida's conclusion? There is no dichotomy between, for example, presence and absence or truth and falsity. Rather, they are intertwined and therefore not fully distinguishable.
I infer that the underlying -- and often implied but never fully stated -- dichotomy which Derrida rejects is A versus non-A. In other words, Derrida's whole project was dedicated to (1) attacking identity in metaphysics (ontology), and (2) promoting skepticism in epistemology.
(For a generally clear treatment of these obscure subjects, see the articles "Post-Structuralism" and "Derrida, Jacques" in the multivolume Routledge Encyclopedia of Philosophy.)
The only work of Derrida that I have read (if such a term is appropriate) is his Of Grammatology (in Gayatri C. Spivak's translation, John Hopkin's University Press, 1976).
Derrida did not merely try to persuade readers that skepticism is the best path to choose, as quite readable skeptics such as Sextus Empiricus (c. 200 CE) had done. (E.g., see Sextus Empiricus: Outlines of Scepticism, translators Julia Annas and Jonathan Barnes.)
Instead, Derrida created skepticism through his writing style itself. It is almost unintelligible, thus leading the usual reader to believe that Derrida's book is important but impossible to understand because that is the way the world is. To increase the obfuscation, Derrida introduces many new terms and old terms with new, barely intelligible meanings -- and then makes understanding them more difficult by not including a glossary or even an index. Example terms (as applied to deconstructing literature) are: erasure, hinge, logocentrism, transcendental signified, and intertextuality.
Of the series of philosophers I have sampled over the last few years, Derrida is the worst, with Kant (1724-1804) second worst, and Kierkegaard (1813-1855) third worst. (Kierkegaard, by the way, was one historical link between Kant and Derrida; it was Kierkegaard who turned Kant's defensive obfuscation into a deliberate "art.")
Philosophical Hell would be studying Derrida for eternity.