What's the difference between communism and anarchism?
I can talk about communism. Karl Marx was originally an economist and philosopher. He thought that the society, since the beginning, is divided into two main castes: the proletarians* (worker who only own their hands to work and their children**) and the "rich people" (the bosses, those who own the production means). He said that there is a fight between these two, because "rich people" ( or bourgeois capitalist) tend to exploit the proletarians. The Industrial Revolution in Europe permited technological progress, we saw the apparition of factories and mass production but the condition of the proletarians became unbearable: their tasks were exhausting, they had to work in the rhythm of machines like machines, they could not educate themself, etc.
The final goal of marxism (and not socialism, because it can mean a lot of things) is to reach the end of this unfair relationship by destroying castes; there would be no State at the end because, for him, it is always led by the bourgeois; people would organized themself alone, electing people for their localities; there we see similarities with anarchism (In the global idea).
But before that, there must be a revolution where will ermege a powerful but temporary State (led by proletarians) that will end the property era and begin the collectivist one: nationalisation (all the private firms belong to the State, the "people"), planning (the system wants results, they fix some objectives, like production rate, and they must meet it at the end of the year), culture and education (the worker, the human being must improve his mind and understand what's around. In fact, communist regimes, like cuba or USSR, have or had a pretty strong school institution). Work is essential in Marx's philosophy, he thought that a human being has to work to develop himself. That's why there is always all this focus on the Worker and the Industry (the good one, not the capitalist one

).
The issue is that communism require a well developed proletarian caste to be effective. Marx didn't imagine Russia as the model, but UK and France where there were thinkers and well organized trade unions. If there is not enough maturity in the proletarian caste, there are important risks of abuses.
Now Socialism: the real socialism is a different approach to marxism. The point is that they are trying to reach equality by reforms in a democratic system (communism is not democratic). There is no revolution. But Socialism is kind of dead ideologically, parties don't refer anymore to marxism because equality and social improvement is difficult to support in a liberal economical context, it is now social democracy (social capitalism, lol)
Personnally, I am kinda socialist (marxist), French stuff you know.
*That's a 19th century concept, but we can universalize it to all the poor workers.
**Except slaves I guess.
Edit: Very interesting stuff, Karl Marx also predicted that there would be a revolution anyway, or a big end at least. I explain myself: as economist he saw that capitalism was a very bad system that tend to create inequalities and pauperization phenomenons that will make it collapse from itself. Who could buy that this insane way of life could last eternally? Some yolo thinkers like Adam Smith?