So, I went ahead and started googling for the "Hegelian Dialectic", and one of the first things I got was this remarkably strange youtube video. At first I liked it because it was quoting the Hitchhiker's Guide to the Galaxy, then I wanted to like it because it started playing really weird techno music while pretending like it would start explaining Hagel. Then my affinity for it sunk like someone who gets caught squeeling on the mob, as it became abundantly clear that this was some weird propaganda video. In fact, there were a number of weird conspiracy videos that came up, all to the effect of "The government was behind 911 (or some such thing), they used the principles of the Hegelian dialecticto fool us all!"
And so, like someone who's just been rickrolled, I recoil away from the disappointment of youtube to see what the rest of the internet has to say. A dialectic is any method of resolving two different points of view, and most of the light introductions I read amounted to describing the Hagelian dialectic as a three part process: some idea, a "thesis", then some contradiction that comes about as one thinks through the implications of this idea, the "antithesis", then a resolution of the two, the "synthesis". At this point I think to myself "Thanks internet, now that you've reworded the question a bit for me, what's the substance here, how does this synthesis work in Hagel's model?"
And after I ask this I look at the internet, and it looks back at me, and it decides to oscillate between continually rephrasing the question and giving me some philosophical gobbldy gook I have trouble understanding right away. So, I push further into the gobbldy gook, clicking my way down the wikipedia rabbit-hole of links, trying seeing if I can suck any substance out, and here's what I got.
Hagel didn't so much describe things in the "thesis-antithesis-synthesis" sense, but instead in terms of "abstract-negation-concrete", wherein the attempt to find truths in pure abstraction leads to some kind of negation, because the very structure of a truth relies on its context in the concrete. For instance think of left and right, if you try to define the truth of pure left-ness and pure right-ness as abstract concepts on their own, the inherent symmetry between the two means your definitions will be indistinguishable, which contradicts the fact that they are antonyms. You have to put them in the context of one and other and some oriented observer to notice a difference between them. The truth of what left-ness and right-ness are depends on their being in a concrete context, any attempt to abstract them cannot preserve this truth.
So already there's some substance to thinking about dialectics, any seeming contradiction we want to resolve, in terms of "abstraction-negation-concrete" instead of the vapid "thesis-antithesis-synthesis", because part of what Hagel is saying is that philosophical truths themselves rely on existing in the concrete, so resolutions can come about when we take something down from the abstract. This is not at all true about other truths, like mathematical ones for example, which can often flourish from the clarity of the abstract realm.
Furthermore, according to Wikipedia, the "essence of what is popularly called the Hagelian Dialectic" is that "the concrete, the synthesis, the absolute, must always pass through the phase of the negative, in the journey to completion, that is, mediation". For instance, the fact that our resolution for right-ness and left-ness depended on describing one as the negative of the other was no coincidence, and comes up in any one of these resolutions (usually in a less obvious way when it comes to more sophisticated ideas than left-ness and right-ness).
So, what is another example of Hagel solving a dialectic by these means? This is one wikipedia provided, which somewhat clarified things for me (though the specifics are a bit confusing):
"One important dialectical principle for Hegel is the transition from quantity to quality, which he terms the Measure. The measure is the qualitative quantum, the quantum is the existence of quantity.
"The identity between quantity and quality, which is found in Measure, is at first only implicit, and not yet explicitly realised. In other words, these two categories, which unite in Measure, each claim an independent authority. On the one hand, the quantitative features of existence may be altered, without affecting its quality. On the other hand, this increase and diminution, immaterial though it be, has its limit, by exceeding which the quality suffers change. [...] But if the quantity present in measure exceeds a certain limit, the quality corresponding to it is also put in abeyance. This however is not a negation of quality altogether, but only of this definite quality, the place of which is at once occupied by another. This process of measure, which appears alternately as a mere change in quantity, and then as a sudden revulsion of quantity into quality, may be envisaged under the figure of a nodal (knotted) line".
As an example, Hegel mentions the states of aggregation of water: "Thus the temperature of water is, in the first place, a point of no consequence in respect of its liquidity: still with the increase or diminution of the temperature of the liquid water, there comes a point where this state of cohesion suffers a qualitative change, and the water is converted into steam or ice". As other examples Hegel mentions the reaching of a point where a single additional grain makes a heap of wheat; or where the bald-tail is produced, if we continue plucking out single hairs."