A new RS translation of §81 in Hegel’s Encyclopedia of the Philosophical Sciences in Outline. 
The dialectical moment is when definite concepts supersede themselves and transition into their opposites. But, especially in philosophy, if the dialectical principle is applied outside of the context of complete understanding, it becomes skepticism, and the result is mere negation.
It’s not uncommon to treat dialectics as mere artifice through which, by playing up the appearance of contradictions, confusion is generated. However, though the juxtaposition can be treated as an illusion, it is in fact real. Yes, the dialectic often presents as nothing more than a subjectivist back-and-forth, a method of swinging between arguments for and against, with superficial ingenuity providing cover for little substance. However, in its true and proper character, the dialectic is the real nature of cognition: the delineation of finite boundaries within a universal continuity.
When we reflect upon a thing we move beyond merely accepting its isolated existence: we give it context by bringing out its relativity, while at the same time retaining its validity in isolation.  Dialectics, then, refers to the immanent march forth, such that the one-sidedness and limitations of existing determinations of positive understanding come to be understood also as negations. Anything that is finite is at the same time superseding itself. The dialectical principle thus constitutes the living soul of scientific progress; it’s the only dynamic which provides immanent coherence and necessity to the discoveries of science, as resulting from an inner movement rather than from external intervention, from beyond what is finite.
Properly understanding and recognizing the dialectic is of the utmost importance. It is the principle of all movement, of all life and all activity in the actual world. The dialectic is also the soul of all truly scientific knowledge.
It is ordinarily considered fair to refrain from subjecting anyone to the rigors of abstract intellectual reason. As the saying goes, “live and let live” — one way is valid, but so is any other. When we look more closely, however, the impositions on any finite thing aren’t merely external; their own nature is the cause of their negation, transforming them into their opposite. For example, we say that man is mortal, and regard dying as something resulting from external circumstances. If this understanding were correct, man would be in possession of two special properties: vitality and mortality. The true view of the matter, however, is that life as such carries within it the germ of death. The finite in general, being radically self-contradictory, entails its own negation. 
On the other hand, dialectics is not to be confused with mere sophistry, the essence of which consists precisely in privileging a one-sided and abstract principle in isolation, as may suit the interest and particular situation of the individual at the time. For example, my actions should be guided by the procurement of the means for my existence. However, if I then single out this aspect — the principle of my well-being — and deduce from it the consequence that I may steal, or that I may betray my fatherland, I am engaging in sophistry. Similarly, it is essential for the subjective dimension of freedom that I have insight into my actions, and the conviction that they are right. However, if I reason on this principle alone, then I slip into sophistry, and throw all principles of morality overboard. 
Dialectics is essentially different from these kinds of reasoning for it aims precisely at considering things in their contextual being and movement, thereby making apparent the finiteness of one-sided determinations of understanding.
Incidentally, dialectics is nothing new in philosophy. Among the ancients, Plato is credited as the inventor of dialectics, and rightly so insofar as it’s in Plato’s philosophy that dialectics first appears in a free, scientific, and thus objective form. In his predecessor, Socrates, the dialectic retained a predominantly subjective character. In accordance with the general character of his philosophizing, it presented as Irony. Socrates directed his dialectic firstly against ordinary consciousness in general, and then against the Sophists in particular. In his conversations he used to give the impression that he wanted to find out more about the matter under discussion; under this pretense he asked all kinds of leading questions, drawing those with whom he conversed into believing the opposite of what they initially had set out to defend. For example, when the Sophists claimed to be teachers, Socrates, through a series of questions, got the Sophist Protagoras to admit that all learning was mere recollection.  Plato’s dialogues are more strictly scientific: he uses the dialectical method to demonstrate that all fixed intellectual determinations are finite. In Parmenides, for example, the One is derived from the Many, only to then demonstrate that the Many is only able to constitute itself as the One. In this grand style did Plato treat the dialectic! In modern times it was Kant, more than any other, who brought the dialectic back to the forefront and restored its pride of place. As we’ve seen, he did this by working out the so-called antinomies of reason.  Far from presenting us conflicts deriving from a variation of grounds and perspective, they are rather a demonstration that every abstract proposition of understanding, taken as a given absolute, turns directly into its opposite.
Howevermuch the dialectic may be difficult to grasp, we must by no means imagine that it exists only as a tool for the philosopher. On the contrary, the dialectic gives expression to what is already found in all degrees of consciousness and in general experience. Everything that surrounds us may be interpreted as the dialectic in motion. We know that everything finite, instead of being fixed and ultimate, is, rather, changeable and transient. This is nothing other than the dialectic of the finite: the same, being other than what it is, turns into its opposite.
Earlier we identified that the idea of the goodness of God was implicit in our ability to understand;  we may analogously remark of the dialectic that, in the same (objective) sense, it corresponds to the idea of the power of God. When we say that all things — everything that we can identify — stands on trial, we express an understanding that the dialectic is a general irresistible power before which nothing, however secure and firm it may first appear, stands aloof. This category does not exhaust the divine essence of the concept of God, but it does form an essential moment of all religious consciousness.
Apart from this general objectivity of dialectic, we find traces of its presence in the domains and phases of the natural and spiritual world. For example, take the motion of celestial bodies. At this moment the planet stands in one spot, but it brings into being the possibility of being otherwise by moving. Similarly, the physical elements prove to be dialectical, and meteorology offers proof of their dialectics. The principle is the same: that which forms the basis of all natural processes is that through which they’re simultaneously driven beyond themselves.
To illustrate the presence of the dialectic in the spiritual world, and even more specifically in the fields of law and morality, it suffices to notice that the principle that an extreme turns into its opposite is recognized in many popular sayings. In law, for example: “Summum ius summa iniuria” — abstract law, taken to its extreme, turns into injustice.  In politics: “The extremes of anarchy and despotism lead to one another.” In individual ethics: “Pride comes before a fall.” “Too much wit outwits itself.” Even feeling — bodily as well as mental — has its dialectic. As is well-known, the extremes of pain and pleasure pass into each other: the overjoyed heart seeks relief in tears, and the most heartfelt melancholy will at times betray its presence by a smile.
Skepticism should not be regarded merely as a doctrine of doubt. It would be more correct to say that the skeptic is absolutely certain of his point: the nullity of everything finite. He who doubts still clings to the possibility that his doubt may be resolved, that one of the certainties among which he wavers will turn out to be solid and true. True skepticism is altogether different: it completely abandons any hope of achieving solidity in understanding, which results in an attitude of imperturbability and self-reliance. This is the case at least of the noble skepticism of antiquity — especially as conceptualized by Sextus Empiricus, who systematized and complemented the dogma of his Stoic and Epicurean predecessors in later Roman times. This lofty ancient skepticism should not be confused with the modern skepticism earlier discussed,  which partly preceded Critical Philosophy, and partly emerges from it. This later skepticism consists solely in denying the possibility of truth beyond mere sensation, insisting that only that within our immediate sense-perception is valid. 
Even today skepticism is regarded as the implacable enemy of all positive knowledge in general, and thus of philosophy, insofar as philosophy is concerned with positive knowledge. But in this belief there is a misconception. It is in fact only the limited thought of abstract intellectualism that has to fear of skepticism, because it is unable to withstand it. Philosophy, meanwhile, entertains the dialectic as a moment within itself — the dialectical moment. However, philosophy goes beyond mere skepticism, unsatisfied with its purely negative result. The skeptic is lulled into error by mere and abstract negation. In contradistinction, the negative result of the dialectic is a positive, for it contains within itself the process of its coming into being.
Thus conceived, the dialectical moment sets the stage for the characterization of the third grade of logical truth: the speculative form, the form of positive reason.
 “The leaves of the tree are green” already establishes a continuity and a discontinuity between the leaf and the tree. Same with “John is a Man.” — R. D.
 If there was no “dying,” the concept of “living” would cease to be useful for communication, or even intelligible. — R. D.
 Domenico Losurdo explores Hegel’s treatment of these categories in section 7.9 of Hegel and the Freedom of Moderns. For Hegel, in the conflict between legal right [Recht] and well-being [Wohl], objective legal right trumps subjective well-being. However, in the conflict between legal right and authentic “extreme need” [Not], this extreme need is constituted as a right unto itself [Notrecht], and emerges supreme as a “sacred term.” According to Losurdo, “Hegel’s position is not characterized by the argument for or the questioning of an alleged right to revolution or resistance (a right that is contradictory in itself), but by the analysis of objective contradictions which, in the absence of timely reforms, make the outbreak of a revolution inevitable.” — A. M.
 See also Nietzsche’s later commentary on Socrates and dialectics in The Twilight of the Idols: “With Socrates Greek taste veers round in favour of dialectics: what actually occurs? In the first place a noble taste is vanquished: with dialectics the mob comes to the top. […] Is the Socratic irony an expression of revolt, of mob resentment? Does Socrates, as a creature suffering under oppression, enjoy his innate ferocity in the knife-thrusts of the syllogism? Does he wreak his revenge on the noblemen he fascinates?” [web] — R. D.
 Lit. “Supreme law, highest injury.” — R. D.