Marxism is a form of socialistic philosophy, economics, and sociology following a dialectical materialist world view. Marxism was invented by Karl Marx and Frederich Engels. Marxist thought primarily consists of solving the issues of the class struggle and how the class system affects socioeconomic development. Marxism has long been an opponent to capitalism and Marxist thought has often been applied to criticise and analyse the develpment of the class system in capitalistic society.
The earliest stage of societal evolution is the primitive hunter-gatherer stage, where the modes of production are nearly non-existent, and people live very modestly. Then, with the advent of agriculture comes either a society based on feudalism or slavery, or both.
In the feudalistic stage, there are a number of classes, including the peasants, aristocracy, and royalty. Then, as nations become industrialized, comes the capitalist stage; this is the stage most countries have evolved to at this point.
In a capitalist society, there are two main classes, the bourgeoisie (capitalist class) and the proletariat (working class). Because the proletariat owns no industry, they are forced to sell their labor to the capitalists. The capitalists are able to live off of the labor of the workers, and they give the workers only as much as they need to.
Eventually, Marxists believe that the proletariat (the majority of the population) will gain class consciousness and realise the ludicrous situation they live in. They will take the means of production from the rich and form a classless, socialist society, where everyone works and benefits from their efforts.
There are several ideologies claming to be marxist. The most prominent of which (and the only ideology to have had successful revolutions) is marxism-leninism. Aside that you also have revisionist marxism, such as Titoism, Trotskyism, Juche and more.
Almost all leftists can give credit to the importance of Marxism in terms of its accurate and thoughtful examination of capitalist and post-capitalist societies.