This might be suited to somewhere more like /r/socialism or even /r/badpolitics. While this question has historical implications (such as how and why these ideologies came to be), they're not historical questions, but rather political science questions.
If you're just looking for a quick, easy-to-digest overview then I recommend this website:
http://www.philosophybasics.com/
It has pages for each political philosophy that you mention:
As the name suggests, this is just a basic starting-point. You might also want to take a look at the entries for Capitalism and Liberalism in order to put the other philosophies into context.
This is a complex question. You can ask elsewhere, like others have suggested, but I'll answer here. I'm usually not smart enough for this sub, but this is my time to shine...or get deleted.
In the history of ideas, contentious terms are always incredibly difficult to pin down. The historical aspect of this question comes in when you start asking about whose socialism/communism/marxism. Since, I have a background in Marxist philosophy I can, at the very least, give you the definitions and conceptual frameworks to these ideas according to Marx and others in the left-Marxist tradition.
Socialism
In Marx and Engel's Socialism: Utopian and Scientific, they explain socialism as the resolution to the class struggle of capitalism.
The proletariat seizes the public power, and by means of this transforms the socialized means of production, slipping from the hands of the bourgeoisie, into public property. By this act, the proletariat frees the means of production from the character of capital they have thus far borne, and gives their socialized character complete freedom to work itself out. Socialized production upon a predetermined plan becomes henceforth possible. The development of production makes the existence of different classes of society thenceforth an anachronism. In proportion as anarchy in social production vanishes, the political authority of the State dies out. Man, at last the master of his own form of social organization, becomes at the same time the lord over Nature, his own master — free.
For Marx, the motor of history is the ability for humans to produce their own means of subsistence, but the exact method and organization of production utterly shapes what society looks like. Marx's work is largely an effort to explain the development of european society through what's called historical materialism. In that section I've quoted above, you'll notice a description of the abstract mode by which capitalism transitions into socialism. The specifics are always up to the "historical conditions" When the workers begin to own the means of production––through revolutionary or other means––it is socialism.
Communism
In Marx's The German Ideology he lays out a definition of Communism that is incredibly different than the way many think of it contemporaneously.
Communism is for us not a state of affairs which is to be established, an ideal to which reality [will] have to adjust itself. We call communism the real movement which abolishes the present state of things. The conditions of this movement result from the premises now in existence.
Here, Marx describes communism as not a "state," but a movement. Communism is not a political economy, but instead we might say that communism is the overall arch of history in a certain direction. There's a materialist metaphysics that underlays Marx that mimics (in a way) Hegel's idealist metaphysics. This probably isn't the time to get too philosophical, so I won't. The point is that communism, according to Marx at least, is not a political economy, but the material process that resolves class conflicts.
Marxism
Marx was an incredibly cantankerous individual. In his letters and writings he comes across pretty eager to pick a fight. See for example, his Critique of the Gotha Program. Throughout this critique he goes through another socialist program line for line and debunks it.
Another great example is his dispute with French socialists and anarchists. Marx himself is probably best known for the statement, "If anything is certain, it is that I myself am not a Marxist." If Marx himself is not a Marxist, then who is? This quote is, funny enough, his response to his son-in-law, Paul Lafargue's, work on the the Programme of the Parti Ouvrier––a French Socialist program. Marx was unhappy with the specific list of demands that arose in Lafargue, and other French socialist's work.
In light of all of this, what is Marxism? I think, at the very least, we might say that Marxism is the intellectual history of other philosophers, historians, political theorists interacting with Marx's thought. Given Marx's rejection of Marxism, it seems wrong to say that Marxism has to do with intellectual purity or agreement, but resonance and affinity. Marxism is about the tendency for other thinkers to grapple with Marx's work.
This is a really complex question and you're sure to get some tough answers, as all of these are really just base ideas with a ton of different implementations. I'll do my best to explain though.
Socialism is the most basic concept behind all these terms. It simply means a system where the workers, or the community as a whole, own the means of production. It is NOT simply any government funded program, nor is it necessarily revolutionary class struggle.
Communism is a form of Socialism partially developed and popularized by Marx and Engels. It advocates revolutionary class struggle by the proletariat (working class) against the bourgeoisie (owners of the means of production in order to establish a socialist system, as well as abolition of private (but not personal) property and a classless society. There are a ridiculous amount of different schools and implementations of communism ranging from Anarchism to Stalinism, mostly based on varying interpretations of Marxist philosophy.
Marxism refers to the philosophical ideas of Karl Marx, which includes Communism but also ideas such as Dialectical Materialism which in turn led Marx to devise Communist theory.
Social Democracy (Different from democratic socialism) refers to liberal democracy which remains capitalist but implements (usually free) government programs such as socialized medicine and education. Think Bernie Sanders or Scandinavia. Democratic Socialism, on the other hand, means an actual socialist society with a democratic government.