What is dialectics? Here are two definitions:
- Dialectics is the theory of change.
- Dialectics is the identity of identity and non-identity: (A=B)=(A≠B)
The first has the problem that it is so vague that really it could mean anything; other theories are also widely considered “theories of change” (e.g. the theory of evolution). The second is rather elegant — certainly unique… but also reads like nonsense. Which one is correct? In my view, both.
Let me explain by way of a personal anecdote. When I went to university I didn’t study philosophy or Marxism. I studied Mechanical Engineering, an engineering focused on how to make planes and robots. One of the classes these engineers have to take is Thermodynamics, and this class focuses on energy and heat. The class is a bit of a stumbling block for many students, and part of the reason is that it introduces the concept of entropy. Entropy is interesting because it is extremely important and ubiquitous, and yet somewhat slippery to even explain. This is what Wikipedia has to say about it:
Entropy is a scientific concept as well as a measurable physical property that is most commonly associated with a state of disorder, randomness, or uncertainty.
If asked to explain what entropy is, students can very loosely and colloquially say it’s “a measure of disorder.” This would be correct, but yet there remains a yawning gap between such an abstract notion (“a shattered vase is more disordered than an intact one!”), and its precise inclusion in formulas that include pressure in Pascals and volume in cubic meters.
I bring entropy up because I think it is important that people don’t get the impression that because dialectics can be defined correctly either in uselessly vague terms or in precise but difficult terms, that this somehow means that users of the term are trying to con or swindle readers. 
Perhaps this seems like it strays very far from orthodox explanations of dialectics, so here’s Lenin referencing Marx and Hegel and Aristotle:
In his Capital, Marx first analyses the simplest, most common, most ordinary and fundamental everyday relation of bourgeois society, a relation encountered billions of times: the exchange of commodities. In this very simple phenomenon, in this “cell” of bourgeois society, his analysis reveals all the contradictions — or the germs of all the contradictions — of modern society. The subsequent exposition shows us the development (both growth and movement) of these contradictions and of this society in the sum of its individual parts, from its beginning to its end. This should be the method of study and presentation of dialectics in general, for Marx’s dialectics of bourgeois society is only a particular case of dialectics. […]
Begin with what is the simplest and most ordinary or common, with any proposition: “the leaves of a tree are green,” “John is a man,” “Fido is a dog,” etc. Already here we have dialectics, as Hegel’s genius recognised: the individual is the universal. […]
[Citing Aristotle] “We could not suppose that there is a house besides the particular houses.” 
Lenin claims that the statement “John is a man” is already charged with contradiction. How? Perhaps, as with entropy, the inclusion of mathematical symbols can assist us in understanding the scandal.
Could we say that “John is a man” translates to “John=Man”? Were it so, either the term “John” or the term “Man” would be rendered superfluous, it would suffice to say “John is John” or “Man is Man” in all circumstances. However, we know we are conveying something non-trivial. We are saying that John belongs to the category of man (we could use the logical symbol ∈). It’s not so simple, though. There’s a philosophical issue here. Recall that phrase of Aristotle cited by Lenin: “We could not suppose that there is a house besides the particular houses.” We cannot suppose there is “a man” besides all the particular men. Our abstract notion of man (e.g., biped, vocal, male, nameless) is historically and logically only cobbled together from certain common aspects of individual men. This abstract man has no concrete reality, he does not exist outside of those particulars. From this point of view, Man came about from John (and others like John), and there is an important sense in which Man is John.
The simple expression “John is a man,” which we all trivially understand, conveys a notion that in reality turns out to be quite complex: The relationship of a particular to a universal, expressed as the identity of an identity [John=Man] and a non-identity [John≠Man] coexisting in contradiction [(John=Man)=(John≠Man)]. Introducing the notion of belonging (∈) as complement to simple being (=) and non-being (≠) allows us to manage this logical contradiction quite well, and express it very easily: [John∈Man]. However, we must not lose sight of the fact that this was already a conceptual dimensional leap, an important innovation. Contradiction here exerted a kind of cognitive evolutionary pressure. Necessity, as they say, is the mother of invention.
Marx says “Capital is labour.” But, how can this be? Capital is the opposite of labour! So there Marx goes, explaining that capital is crystallized labour, via the commodity cycle. As it turns out, the mere expression of a contradictory relationship carries the implication that there exists a process whereby one can reach the other. This is how the focus on contradiction is deeply tied to change.
The word “change” is apt because really there are very many ways that one concept may turn into its opposite. Perspective determines whether we are still or in motion. Space determines whether a gesture is rude or polite. With time we see the living become the dying, and the dying give birth to the living.
Perhaps this sounds somewhat precious, relying on some juggling on my part. Let us take a step back, and think for a moment about where in our lives we might hear expressions that do not take this contradictory form. I can think of a few:
- “Orders are orders.”
- “It is what it is.”
- “Family is family.”
- “I am what I am.”
- “Freedom is freedom.”
- “Men are men, women are women.”
- “War is war.”
These are all argument-enders. What all of these phrases have in common is that they have an “atomic” quality, where by atomic we mean tight and self-contained. Tautologies.  They’re disconnected from the world in a very important sense. This is to say, they are not describing the world, they are asserting the world, demanding that you accept a finality. There’s no argument to be had, we have simply a premise — fixed, unchanging, universal across time and space — being insisted upon.
Nietzsche, the ultra-reactionary philosopher who mourned the tragedy of the French Revolution, spoke for all aristocratic classes everywhere when he stated:
Socrates belonged, in his origins, to the lowest orders: Socrates was rabble. One knows, one sees for oneself, how ugly he was. […] With Socrates Greek taste undergoes a change in favour of dialectics: what is really happening when that happens? It is above all the defeat of a nobler taste; with dialectics the rabble gets on top. 
Another big enemy of dialectics is the reactionary British writer George Orwell, who in 1984 famously mocks the idea of embracing contradiction as a rhetorical tool for psychological manipulation:
War is Peace
Freedom is Slavery
Ignorance is Strength
The protagonist of the novel, innocent Englishman Winston Smith, is tormented by this establishment propaganda. This establishment is represented by O’Brien, an evil and vengeful Irishman who has usurped his way into power. The world is upside-down!
Aristocrats and conservatives despise dialectics because they feel their privileges threatened by the very notion of change. They covet a static world, which makes them heavily inclined in favor of undialectical relationships. A Roman might say “Masters are masters, slaves are slaves” in the same way modern propaganda will suppress questioning the veracity of a claim by insisting “Evil is evil.” This is an injunction for one to accept what they are being given.
Rome immediately invites one to think of Christianity, which is long-lived enough that it not only offers examples of dialectical and non-dialectical thinking in its doctrine, but itself embodies the notion of a concept turning into its opposite. Roman society was ruthlessly hierarchical, with rigid hereditary castes, and their religion of eternal and whimsical gods was fit for their purposes, legitimizing a society where men did as they wished with their slaves. Though today Christianity is largely a hegemonic and reactionary force, once upon a time it was considered deeply revolutionary, and the Romans aggressively attempted to stamp it down.  This is because once upon a time Christianity’s creed was considered explosively dangerous on account of its dialectical proclamations (“The meek shall inherit the Earth!” “God is Man!”). Today, however, Christianity is in power, and thus its rhetoric aims to preserve the status quo, with an emphasis on absolute obedience and non-questioning (“God is God!” “Marriage is marriage!”). In the past Christianity was an agent of change, today it is an enemy of change.
The aforementioned examples were chosen to illustrate the stark antagonism between the expressions of the likes of Marx and the likes of Nietzsche, but we must be aware that anti-dialectical thinking in everyday life can take subtler forms.
In his essay on dialectics Lenin defends the theory of “development as a unity of opposites” against those who concede change, but theorize it as mere part of “a cycle of increase and decrease.”  After all, change is observable, and it is rare for someone to be so obtuse so as to deny it outright. However, anti-dialecticians have another trick up their sleeve: a jaded and cynical attitude towards history as a “cyclical slaughterhouse.” They concede that, yes, things change, but they change according to the whims of fate, karma, accidents, willpower, or God; in the end, in their view, everything is a wash; things never really change.
This is the reason why Lenin argues that it is essential to grasp the source of change: the struggle of mutually exclusive opposites. If we do not, we end up among the ranks of those who are clueless about how to effect change. Lenin polemicizes not only against powerful aristocrats and reactionaries, but also against all sorts of sentimental idealists (anarchists, social democrats, left-communists) who desire a better society but don’t particularly care for or aren’t careful about the details or the process of how to achieve it:
Despair is typical of those who do not understand the causes of evil, see no way out, and are incapable of struggle. The modern industrial proletariat does not belong to the category of such classes. 
In short, he rails against those who prove incapable of carrying out “a concrete analysis of a concrete situation.”  Similarly, Marx and Engels’ polemics not only target bourgeois economists, but also those within the ranks of “the left”: anarchists like Proudhon and Bakunin and right-opportunists like Dühring.
The virtue of concrete analysis is expressed in strategic creativity that at once stands out from both resignation and utopianism. In Marx and Engels the championing of the proletariat above the peasantry or lumpenproletariat; in Lenin the Party as vanguard, the Brest-Litovsk Treaty, and the NEP; in Stalin timely collectivization and tactical diplomacy that draws gleefully spectating Western powers begrudgingly into war; in Mao a pan-Chinese alliance against Japanese occupation, led by a Peasant Army but including even capitalists in its ranks; in Deng irrevocable integration with the incipiently hegemonic capitalist order coupled with a fierce defense of the autonomy of the political executive; etc. This isn’t a list of revolutionary “Greatest Hits.” I focus on controversial decisions and achievements because this is precisely where we see most clearly how dialectics is not a question of cleverer rhetoric or better poetry, it’s an approach that produces creative strategy that seizes victory from the jaws of defeat again and again.
In short, it is absolutely essential to grasp not only that relationships will develop, but also that they develop because of the struggle of mutually contradictory opposites — that the contradiction at the core of their self-identity is inescapable. This implies victories are never final: “When a contradiction is resolved, new contradictions emerge, and competition takes place again. In this way, society constantly progresses.”  This leads to a kind of pessimisitic optimism: opportunities will necessarily continue to arise, and therefore we need to be organized and prepared to consistently seize these opportunities in order to effect our desired outcomes. This is the difference between rebellion and revolution.
So, there we have it. As we can see, dialectics is a rather expansive concept. It is at once a logical process, a philosophical tradition, a manner of speaking… frankly, it’s an entire way to approach reality. If this sounds scandalous, do remember that we speak of people being “logical” in a way that is at least analogous.
Dialectics posits that mutually exclusive opposites lie at the very essence of all relationships, that this contradiction is the driver of the development of these relationships, that everything is ultimately a relationship, and that the only absolute is change.
Some say that you can understand Marx fine without grasping dialectics, but Marxism without dialectics is a boat riddled with holes. Marxism is often mischaracterized as set of conditions that define utopia, but in reality it’s a theory of how existing capitalist society works and of how to change it. Dialectics is thus not about “accepting change,” or about “hoping for change,” or about “embracing change,” but about really drilling down and understanding why and how change comes about, so that we get comfortable with it and are able to consciously pursue more of it.
This understanding also reveals that certain specific changes are inevitable. Dialectics therefore turns out to be the sobering, bracing antidote to both cynical nihilist resignation and immature utopian idealism. It dismantles the elitist complaint that “human nature is hopeless” and that the masses are vulgar and inept, and it also dismantles its dialectical pair of frenetic and irrational insistence that faith can overcome any challenge through “will to power.” These are both replaced by serene strategizing. We can continue speaking of optimism and faith, and rousing emotional speeches still have their place, but organizing should never feel like a “Hail Mary pass.”  Existing systems are designed to handle such last ditch efforts with minimal disruption. Dialectics imbues even small strategic acts with a long-term sense of purpose and safeguards organizing from the lure of theatric “activism,” and this consistent and steady action yields results.
Grasping dialectics won’t grant any victories in and of itself. In my case it felt more like the crystallization of notions I was already (always?) hazily aware of. However, the way I analyze situations now tends to lead to optimism — a hard-headed kind of optimism that bears little relation to a wish upon a star. I hope others find this explanation similarly encouraging.
It’s possible in fact that there’s a philosophical connection between entropy and dialectics. Both have been described very abstractly as “the arrow of time”! Exploring this connection, however, is not what this essay is about. The point is only that valid definitions of varying concreteness coexisting doesn’t take away from the reality of the underlying concept. This is the case not just for entropy and dialectics, but also for terms like love, game, and sport. ↩
Of course tautologies can themselves be understood as illustrating the principle of dialectics; even though they have the form A=A, that they are understood to be saying something beyond mere repetition derives from the fact that the second term is not exactly the same as the first. ↩
Friedrich Nietzsche, 1888. The Twilight of the Idols. ↩
A very long forward pass in American football, typically made in desperation, with an exceptionally small chance of achieving a completion. ↩