Lots of great answers here. My way to explain it would be the following: materialism is understanding that human society flows from the material realities of history (summing up the climatic and agricultural conditions, the political machinations, the level of technology and resources that are directly available or available through trade, etc. etc.); dialectics is a method of identifying a resolution to two apparently contradictory truths.

So in Hegel, a classic example of the dialectics is trying to understand the transition between ice and water. In terms of temperature, we can observe that at one temperature the actual quality of the water changes from liquid to solid ice. The quantitative change in temperature eventually becomes a quantitative change in the material. And today, we're all familiar with a resolution to this, which is the molecular behavior of water molecules. The part of this that is dialectics is in noticing that one concept of water (liquid water) is contradicted by another concept of water (ice).

Putting the two together, what Marx was trying to do was understand Capitalism and what its actual motive forces are. Unlike the physics example above, you can't use the same sort of direct experiments in a laboratory setting to build more correct models of the means of production and the nature of capitalism. What Marx comes up with that is a great example of historical materialism is his notion of value. Marx spent much of the time working on Capital finding all the evidence that he presents in later volumes and chapters. However, he decided to start with his theory and then present the evidence, so the most important dialectical materialism of the book is actually right in the front. Here's the contradiction in what makes things valuable under capitalism: on the one hand, everything generally has some kind of use value that is qualitatively its purpose -- food is produced to be eaten, CPUs are produced to play Netflix, a coat is made to be worn in the cold -- but on the other hand, specifically under the social formation that could be identified as Capitalism, all things have an exchange value, which is to say you can always sell it for some amount of money and then exchange that money for something else with a use value, which of course you will inevitably do. The materialist solution to this dialectical contradiction in what it means for a product of someone's labor to be valuable is to understand value as a more abstract concept than either use value or exchange value, and identify it as being equal to the socially necessary labor time to produce the thing. If this is a good resolution to this materialist dialectic, then we should be able to connect both use value and exchange value through this further idea of value as socially necessary labor time. We can of course, do that. Technological advances make some labor less valuable because it simply takes less time. We can see from this vantage point as well that production can only be reasoned about in terms of the only input we can actually change: how much time a person spends doing a task. Since that's all that matters, we can see that the use value is accounted for insofar as people's needs and wants have an optimal amount of socially necessary labor time to meet, and that as technology improves, the real value of something whose production is made more efficient is obviously lesser. A machine produced coat lessens the value of coats because it simply takes one less time to produce it. And there isn't anything socially necessary about a cottage industry of coat-making over industrial factory coat-making, and thus the cottage worker finds their skills much less valuable than they used to be. Value in exchange, which is to say the price of goods, will change based somewhat on supply and demand, but the price of labor will depend primarily on how good Capitalists get at extracting from workers. And at a certain point, there it is. Capitalism is class war, built right into a basic resolution to what it means to do wage slavery and sell goods on "the market."

I'm probably wrong in some critical way and I'll get dunked on by the true Marx readers, but I think I'm correct about explaining the idea of the materialist dialectic.