Marxism contains the philosophy of dialectical materialism, which basically states that only the material world exists and all things in are generally in a state of change and development from one stage to the next and primarily develop according to internal contradictions (opposing forces) within the things themselves. Marxism also applies this philosophy to analysis of human society and history, and looks at the reasons that societies change over time. The main contradiction in class-based human societies is the class struggle, which leads to development of the society. The class structure is determined by the economic system of production and technology, and as those change so too do relations between the classes. Marxism then claims that the class struggle under capitalism will lead into socialism due to the fact that the development of capitalism and the continued mechanization of labor will lead to greater and greater antagonism between the two main classes of capitalism, up to the point where there is a social revolution and a socialist society and state (in contrast to anarchism) is established. Socialism is a society in which there is a state run by the working class which is used to fight against and destroy the capitalist class. Once the capitalist class is destroyed, then the working class will have liberated itself and then there will be no economic classes in society, and then the state's functions can be adapted into functions of the people in general and they state-as-such can wither away and there can be a stateless classless society where all property is held in common, which is the ultimate goal of Marxism.
https://www.marxists.org/archive/lenin/works/1913/mar/x01.htmAs to the labels:
>MarxismWhat I described.
>Marxism-LeninismLenin added some further analysis onto the basic Marxism because of further developments of capitalism that occured after Marx's time. Mainly the development of capitalism-imperialism and the transformation of free-competition capitalism into monopoly capitalism. Lenin also outlined the need for a communist vanguard party to lead the workers. It's said that Leninism is the Marxism of the era of imperialism and socialist revolution.
>TrotskyismTrotsky claimed that the Soviet Union didn't turn out well due to political errors. Nowadays it basically means you think there should be "more democracy" in communism than there was in the USSR. As to what that means, it varies.
>Marxism-LeninismThis is what Trotskyists call people who disagree with their "democratic" ideas.
>Maoism or Marxism-Leninism-MaoismFurther development of Marxism-Leninism to better address the need to avoid capitalist restoration within a communist society. Mainly the recognition that the class struggle continues all the way during socialism, until communism is fully reached. This wasn't fully grasped by the USSR.
Everything else is pretty much special snowflakeism.