dialectical materialism is, in a very condensed and simplified summary:
dialectics: a form of logic originating in German philosophy (Marx's predecessors); it's a way of conceptualizing and understanding the world, centered on change; the world is constantly changing because of an innumerable number of social/economic/political forces acting upon it, and with the dialectical framework you seek to try to identify these forces and how they interact, and how their interaction creates change, if you've heard of thesis-antithesis-synthesis, that's a simple statement of it: a force, an opposing force, and the result of the clash of these forces; of course you cannot identify every single force of change and you may incorrectly identify them, so one seeks evidence via history and current events; essentially it's a way of trying to understand how the world works, it's like a formulation of something akin to the scientific method to social sciences; you observe, create hypotheses, test hypotheses, then you create falsifiable theories from your results
because the dialectical method places a strong emphasis on creating testable hypotheses and seeking evidence to create falsifiable theories, it necessarily requires these hypotheses to be strongly grounded in the material world and history to draw valid, applicable conclusions that are useful; the theory is just a way of trying to explain the world as it is, but the world and the world's change exists prior to the theory; theory is a human/philosophical invention to try to understand the world; if a theory is proven wrong by history and current events you should reject it, understand why it is wrong and try to revise and adapt it to new evidence
dialectical materialism is, in the shortest summary: an adaptation/formulation of the scientific method to social sciences
I would avoid using thesis-antithesis-synthesis framework, it is ok for discussions among people who already understand dialectics, but it's a very bad tool for learning. For example, the word synthesis gives an impression of the opposing forces somehow combining or merging, when in reality the case is often that one of them destroys the other. The contradiction between the bourgeoisie and the aristocracy did not lead to a merging, it was an often violent overthrow. But it was synthesis in the sense that the resolution of the contradiction demanded and resulted in changes in the bourgeoisie, and the new bourgeoisie born out of it can not be understood without understanding the now dead aristocracy.
I might also mention the change of quantity into quality and negation of the negation, even if they might be less important than contradiction.