Lots of people has great answers here. I would love to explain a bit of the reasoning behind these production relations in very crude and hand-wavy terms.

At the time of Marx, people see the economy consists of two major inputs, labour and means of production (land, machinary, tools, etc), with machinary and tools (things that can be produced and enables production) called capitial.

In a capitalistic society, like its name suggest, capital is a valuable resource. That means people owning these capital, aka capitalist, can make money just by renting these capitals without contributing labor.

However with the productivity increase, it was theorized that capitals will lose its value, hence enabling workers to collectively own the means of production. At that stage, only labour will be valuable resource, hence the compensation will be directly tied to the value of labour a worker can provide.

Finally, when the productivity is way over the capacity of consumption, communism is achieved. Human will no longer fight for resource, since the amount of resource can fullfill the need of any individual. Thus the society will be able to distribute resource simply by need.

However like other has said, there are many means of achieving these ideals, not necessarily by pure market changes or by a authoritarian state.