If the Dialectic of Enlightenment is the diagnosis, Negative Dialectics (1966) is Adorno's attempt at a philosophical response. It is one of the hardest books in the philosophical canon, and deliberately so. Adorno thought that clarity was a form of complicity: a text that could be easily summarized had already accommodated itself to the demand for comfortable, consumable knowledge.
His target is what he calls identity thinking: the tendency of reason to subsume the particular under a general concept, to say that this specific thing is an instance of that general category, and to treat the subsumption as adequate knowledge. All conceptual thought does some of this: a concept works by abstracting away from the particulars of individual cases to identify what they have in common. Adorno's point is that this abstracting-away always involves a remainder: the non-identical, the aspects of the particular that the concept cannot capture and that resist being subsumed.
Here is a concrete example. The concept "human being" picks out something real about individual people. But it abstracts away from what is irreducibly particular about any specific person: their specific history, their specific face, their specific way of suffering and hoping and dying. When bureaucratic or political systems apply the concept "human being" (or worse, the concept "citizen," "enemy," "Jew") to actual individuals, they operate on the abstracted concept and treat whatever the concept does not capture as irrelevant. Administrative violence, Adorno thought, operates through the triumph of the concept over the particular. The Holocaust was a systematic attempt to reduce particular human beings to instances of a category and then to eliminate the category.
Negative dialectics is the philosophical practice of resisting this reduction: of constantly attending to the non-identical, to what in any particular thing exceeds and escapes any concept applied to it. This is why Adorno writes the way he writes. He does not want to produce a philosophical system, because a system is the conceptual version of the same domination he is criticizing: it subsumes everything into its framework and claims to have achieved total knowledge. He wants philosophy to maintain what he calls the force of the negative: the critical capacity to say that things are not what they present themselves as, that the current state of the world is not the necessary or natural state of things, that what exists is not what ought to exist.
The phrase that captures his political and philosophical orientation: "To write poetry after Auschwitz is barbaric." This line is routinely misread as a claim that art should stop. What Adorno meant is that art must reckon with the fact that the cultural tradition in which it participates produced the civilization that produced Auschwitz. Art that does not carry that reckoning, that presents itself as beautiful in the old confident way, is an obscenity. But art that does carry it, like the late Beckett plays Adorno loved, is not just possible but necessary: the only honest form of cultural activity in a world that has demonstrated what it is capable of.