Along with the heading, it would be much appreciated if you paste useful links that go in depth of the differences. For this forums scope, I only need to hear the basic difference between these three political structures. I wouldn't mind if you elaborate.
Take a look at this, gives a fair summary... https://www.youtube.com/watch?v=lRbxmQv8Iyk
Socialism = The State owns the means of production (rather than private companies), and distributes the result
Communism = No State/Government nor any private companies. The People themselves decides communally what they need and how to best use resources. Has never been done!
The idea was that Socialism would replace Capitalism, and that Communism would replace Socialism. But while Communism may initially have been the "final goal", "communist" states like the Soviet Union, remained stuck in Socialism (probably because "power corrupts, and absolute power corrupts absolutely").
Of course, many means "social-democracy" when they talk about "socialism"; I.e. a system where there are private companies, but where the Government provide lots of services (like school and healthcare) and perhaps owns companies for key infrastructure (e.g. telecommunication, train&rails, and radio/TV).
Marxism was Carl Marx's theories about communism and how it could be achieved... as opposed to Leninism, Stalinism and Maoism - which all may have had more or less the same goal, but differed in how to get there.
4+1 for adding social democracy to the list - a word which appears to be absent from the vocabulary of many US politicians.– Philipp ♦Aug 26, 2015 at 13:55
Marxism has two meanings:
Any theory of society and economics that is primarily derived (and/or extended) from Karl Marx's theories of society and economics.
Any political platform designed to achieve the revolution Marx predicted, and/or implement the utopia that Marx proposed that revolutionaries work toward.
Marx supported political parties that called themselves "Communist". These "Communist" parties considered themselves to be a sub-set of the "Socialist" parties.
For further information, you can read some of Marx and Engels' writings, including Das Kapital (which presents an economic theory) and The Communist Manifesto (which was the political platform of the German Communist party during the revolution of 1848).
In later generations:
Lenin, Trotsky, and Stalin each claimed to extend Marx' theory and platform. The resulting ideologies are often called "Marxist-Leninism" or "International Socialism"; the "Communist International" or "Comintern" was an organization that Stalin controlled that coordinated the activities of many of the world's communist parties (and some of the world's labor unions).
Mao also claimed to extend Marx' and Lenin's theories and platforms. The resulting ideologies and guerrilla war practices are often called "Maoism".
Most professional economists consider Marx' work on economic theory to be a substantial part of economic theory. Marx and Engels are included in the list of great classical economists, along with Adam Smith, Jeremy Bentham, David Ricardo, and Thomas Malthus. Marx pointed out that the long-term equilibrium for the "standard of living" is what a class insists upon before having, raising, and equipping enough children to replace itself. Several key concepts of economics (including "Cobb-Douglas production functions" and "value added") are designed to explain Marx' observations about the fraction of a factory's revenue that is not paid out for materials or proletarian labor.
Many political parties whose platforms include prominent features of The Communist Manifesto and Lenin's reforms seek to disassociate themselves from the totalitarian reputation of Marxist-Leninism. Many of these parties call themselves "Socialist" or "Progressive".