The most simplistic explanation I've ever heard is that you start with a thesis that explains some phenomenon. Then, an antithesis arises which disproves your thesis (like an exception to the rule) - this is the contradiction, the thesis and antithesis are now in conflict. Thesis and antithesis struggle against each other (or, you argue and work with someone of an opposing opinion) until a new explanation is created called the synthesis which grows out of the thesis but takes into account the antithesis. The synthesis acts as the new thesis now and necessarily contains its own contradiction, and it will further progress into a new synthesis once it comes into contact with its antithesis.

I've also seen the thesis-antithesis-synthesis trio also termed A-not A-B or the abstract-negative-concrete. Also I think the whole thesis-antithesis-synthesis explanation is a bit inadequate if you are really trying to understand dialectics. But for just a simple, general explanation that can help you to generally understand people when they mention it, thesis-antithesis-synthesis I believe is okay.

For the Marxist context, which I assume is why you are asking, Marx applied dialectics to the material world, economic systems resolving themselves into higher economic systems (slave-based->feudalism->capitalism) through the class struggle (the contradiction). Like, in capitalism, the bourgeoisie (the thesis) cannot exist without the proletariat (the antithesis). And so the capitalist system relies on a proletarian class, yet the proletarian class will eventually cause the destruction of capitalism (the thesis creates its own antithesis). So there is a contradiction between the bourgeoisie and proletariat (class struggle) and it will resolve itself into socialism.

Thats a really vulgar explanation of dialectical materialism but I believe it will get you by