Marxism is the adherence to the economic and political theories of Karl Marx (and Friedrich Engels), as expressed in writings such as Das Kapital (Capital), The Communist Manifesto and countless pamphlets.
Marxism is based on the Hegelian idea of Dialectic, specifically Social Dialectic, and envisages that socialism and eventually communism are inevitable replacements of capitalism, which is just a transitional phase of society.
Marxist thinking has never been fully put into practice, although various mutations have, from Social Democracy, through to Marxism-Leninism and Maoism. Most Marxists these days are not dogmatic, but use Marx's ideas as an inspiration for social change.