Derrida's diagnosis of Western philosophy is gathered under the term logocentrism, a term he builds from the Greek logos, which means both word/speech and reason/truth. The claim: Western philosophy has systematically organized itself around the privilege of presence, the idea that genuine meaning, genuine truth, genuine being are all characterized by being fully present to themselves and to a knowing subject. The present moment is more real than the past or future. The spoken word (uttered by a present speaker to a present listener, immediately available for correction and clarification) is more genuine than the written word (absent from its author, available to be read by anyone in any context, unable to defend itself). The original is more authentic than the copy. The inside is more genuine than the outside.
Derrida calls this cluster of related preferences the metaphysics of presence, and he traces it through the history of Western philosophy with remarkable consistency. Plato distrusted writing, the Phaedrus has Socrates argue that writing is a dangerous supplement to genuine memory and living dialogue. Rousseau, in his Essay on the Origin of Languages, treats writing as a corruption of the pure immediacy of spoken language. Saussure, in the Course in General Linguistics, treats the spoken language as the proper object of linguistics and writing as its external representation. Husserl's phenomenology aims at the pure presence of experience to consciousness. Lévi-Strauss treats pre-literate oral cultures as somehow more natural and pure than literate ones. In each case, a hierarchy is being established: presence/speech/origin/inside is privileged over absence/writing/supplement/outside.
Derrida's deconstructive move is not to reverse these hierarchies, not to say that writing is better than speech, or that copies are better than originals. It is to show, through patient close reading, that the hierarchies cannot be sustained on their own terms: that the texts which privilege presence over absence inevitably rely on what they demote. That speech already has the structure of writing (it is repeatable, citable, separable from its original context, these are not accidental has but constitutive ones). That the original already contains the supplement. That presence is already contaminated by the trace of absence.
The concept of the trace is central here. Every element in a sign system carries within it the traces of all the elements it is not, it is defined by its differences from absent terms. This means that presence is never pure: it always already contains the traces of what it is not. There is no pristine origin, no pure self-present meaning, no uncontaminated beginning. The supplement, what is supposed to be added on from outside, to be merely secondary and derivative, turns out to be constitutive. Rousseau thinks writing supplements and corrupts pure speech; Derrida shows that the structure of language Rousseau describes is already supplementary through and through, already requires the iterability and difference that he attributes only to writing.
The phrase that most provoked Derrida's critics: "Il n'y a pas de hors-texte", usually translated as "there is nothing outside the text." This became the rallying cry for accusations of idealism and relativism: Derrida is saying the real world does not exist! But what the phrase actually means, in context, is something more specific: there is no position outside the differential structure of textuality and signification from which we can access a pure, unmediated, pre-interpreted reality. This is not the claim that trees and mountains do not exist. It is the claim that our access to them is always mediated by the differential structures of language, interpretation, and context, that there is no experience that comes to us uncontaminated by the sign-structures through which we encounter and organize the world.