Karl Marx was a philosopher and economist. He wanted to understand class relations and social conflict, so he developed theories to explain why things are the way they are. A Marxist uses Marx's theories to understand why the world is the way it is.
Marx had a lot of theories, such as historical materialism - that all history was primarily motivated by socio-economic forces, not supernatural forces or grand conspiracy. Marx wrote that the dominate socio-economic system running the world in his time was capitalism/imperialism which fueled capital accumulation through exploitation and alienation, and used technology to further this process with imperialist wars for resources etc... He also focused on class struggle between those with the most resources, and those with the fewest resources - the bourgeoisie (capitalists) vs. the proletariat (workers/peasants).
Marx went further than trying to explain why the world is the way it is, he also theorized on how humanity could replace the dominate socio-economic system, and what a non-exploitative non-alienating socio-economic system might look like. "Marxist" refers to anyone who believes Marx's theories are valid and uses them to understand the way things are.