To put it bluntly, what killed 20th century Socialism was the fact that the so-called first stage of communism, which was expressed as Socialism couldn't evolve into the second stage. Now, the reasons behind this tragic failure are the material conditions, doubtlessly, although remnants from the 20th century socialists still blame each other. Yet the important thing to do is trying to understand how the first stage of communism will look like and how it will differ from 20th century socialism.

The first thing that comes to ones mind while trying to make a brief description attempt is to look at what Marx thinks about the subject. Yet Marx did not write a book or even an essay on what communism will look like. This is pretty understandable, considering the fact that Marx lived in 1800s and we still live under capitalism in 2000s. If someone thinking like Marx lived in ancient Greece, that person would predict that the capitalist system would emerge when the material conditions made it possible, but that person could not say how it would be, or when it would be because that person would not have any idea about the variables, about the way historical conditions will make capitalism possible. The same thing happened in 1800s, Marx was able to tell us that communism was going to come, but he wasn't able to tell how or when. Even now, 150 years after the publishing of the Communist Manifesto, we can't exactly tell how it will be like, or in fact when it will be like. Yet, when we look at Marx's works, we can find traces on his thoughts on the communist society of the future. Let's get one thing clear, Marx certainly did hope to see a successful revolution in his lifetime, and the Paris Commune definitely made his hopes seem even more possible to him. Probably this is the reason why he wrote what he wrote on the communist society of the future, and if he lived long enough, he was planning to sum up his vision of communism in the end of Das Kapital.
So let's see what we've got here:

In the Communist Manifesto, Marx demands: "1) Abolition of property in land and application of all rents of land to public purposes. 2) A heavy progressive or graduated income tax. 3) Abolition of all right of inheritance. 4) Confiscation of the property of all emigrants and rebels. 5) Centralization of credit in the hands of the state, by means of a national bank with state capital and an exclusive monopoly. 6) Centralization of communication and transport in the hands of the state. 7) Extension of factories and instruments of production owned by the state, the bringing in cultivation of waste lands, and the improvement of the soil generally in accordance with a common plan. 8) Equal liability of all to labour. Establishment of industrial armies, especially for agriculture. 9) Combination of agriculture with manufacturing industries; gradual abolition of the distinction between town and country, by a more equable distribution of population over the country. 10) Free education for all children in public schools. Abolition of children's factory labour in its present form. Combination of education with industrial production, etc., etc."
Those 'demands' Marx makes are usually misunderstood because they are seen as reforms to transform the capitalist society. They are not! Those ten demands are written to be instant measures taken after the revolution instead of slow reforms following it, so Marx's demands are the material results of the revolution. Here, Marx thinks that the masses will take control of the state and use it to take those measures, but after the fall of the Paris Commune, Marx changes his mind on this and decides that “the working class cannot simply lay hold of ready-made state machinery, and wield it for its own purposes.” So what this means is that we will read Marx's demands regarding the post-revolutionary period by replacing the word 'state' with the word 'organized workers' or 'united proletariat' etc. After all, the only organism that remains is the organization of the workers society itself. So it reads like this: the organized workers will rapidly start abolishing private property and control all the public services. So with this replacement, we see that what Marx demanded us to do, from the 19th century, is in fact very close to our vision of communism.

Marx describes the first stage of communism as the 'dictatorship of the proletariat'. In the 'Critique of the Gotha Program' he says "Between capitalist and communist society there lies the period of the revolutionary transformation of the one into the other. Corresponding to this is also a political transition period in which the state can be nothing but the revolutionary dictatorship of the proletariat." The word 'revolutionary' is extremely important here. It does not only imply that the transformation will be a revolutionary one, but it also implies that the transformation will occur during the process of revolution, which doesn't end until private property is abolished. After the very moment of revolution there is nothing stopping the workers from abolishing private property. In facts, the seeds will have been thrown even before the moment of revolution, at the beginning of the process of revolution. When the process of revolution is over, revolution will be in permanence, or to phase it a clearer way, revolutionary evolution will occur. Marx explains this in 1844 Manuscripts: "It takes actual communist action to abolish actual private property. History will lead to it; and this movement, which in theory we already know to be a self-transcending movement, will constitute in actual fact a very rough and protracted process. But we must regard it as a real advance to have at the outset gained a consciousness of the limited character as well as of the goal of this historical movement –and a consciousness which reaches out beyond it."
Now, if we return to another aspect of the 'dictatorship of the proletariat', which is in fact a very obvious one, we see that it is a form of dictatorship enforced by the proletariat collectively. So it is not only a dictatorship of the proletariat, but it is also a dictatorship by the proletariat and a dictatorship for the proletariat. In fact the whole concept of the 'dictatorship of the proletariat' was developed against Blanquists who supported a ruling elite that was to work for the 'best interests' of the proletariat.

Now, let's see what Marx thinks about Socialism. He writes in 1844 Manuscripts "Socialism is man's positive self-consciousness, no longer mediated through the annulment of religion, just as real life is man's positive reality, no longer mediated through the annulment of private property, through communism. . . . Communism is the . . . actual phase necessary for the next stage of historical development in the process of human emancipation. . . .Communism is the necessary pattern and dynamic principle of the immediate future, but communism as such is not the goal of human development - which goal is the structure of human society." So what Marx means is that Communism is an "actual phase" of society prior to the next historical stage of human development, whereas socialism is "man's positive self-consciousness." Therefore according to Marx, socialism is the goal, it is the intellectual utopia itself. In a way, we can even call it the next step in the biological evolution of human mind. Now, there is a nice play Marx makes with words. He is well known for calling self-proclaimed socialists of his era 'utopians'. Here, he calls the intellectual utopia 'socialism'.

Now, lets look at what the leading 20th century socialist, Lenin, thinks about this whole process of the transformation, the dictatorship of the proletariat etc. In his State and Revolution Lenin says "The dictatorship of the proletariat, i.e., the organization of the vanguard of the oppressed as the ruling class for the purpose of suppressing the oppressors, cannot result merely in an expansion of democracy. Simultaneously with an immense expansion of democracy, which for the first time becomes democracy for the poor, democracy for the people, and not democracy for the money-bags, the dictatorship of the proletariat imposes a series of restrictions on the freedom of the oppressors, the exploiters, the capitalists." Lenin's definition of the 'dictatorship of the proletariat' is directly contrary to the purpose Marx had while inventing the term. Lenin sincerely agrees with Marx on what needs to be done, but he thinks this should be done by an elite vanguard organization. The roots of Lenin's ideas of vanguard are not the subject of this topic, but Lenin is quite sincere about the vanguard as well, which actually shows how naive he was on class relationships. Lenin makes one of his greatest theoretical mistakes when he completely misunderstands Marx's Critique of the Gotha Programme. In State and Revolution, he states that "the social order termed by Marx the first phase of communism" was "usually called socialism" and he supports this claim by reducing the first stage of communism to: 1. common ownership of the means of production; 2. "from each according to his ability, and to each according to his labor"; 3. "the distribution of products" is not yet equal. All three concepts differ from counterparts in Marx's Critique of the Gotha Program. First, Marx specified that the means of production must be owned by "the association of producers" whereas Lenin's imprecise term "common ownership" has consistently been interpreted to mean state ownership -- something that Marx and Engels rejected out of hand. Second, Marx's formulation is "from each according to his ability, and to each according to his labor power." This is quite different from Lenin's "to each according to his labor": Marx's "labor power" is measured solely by the length and intensity of labor, whereas Lenin's "labor" is measured by the value of the commodities it produces. According to Marx, payment for labor is the hallmark of capitalism, whereas payment for labor power is the hallmark of the first stage of communism. Therefore the "socialist" society Lenin described has a basic capitalist feature because it pays wages for labor. Third, Marx was critical of the socialist idea, repeated by Lenin, that distribution of products is central; Marx's view was that conditions of production are more essential, and it is incorrect to "make a fuss" about distribution: "The distribution of the means of consumption at any time is only a consequence of the distribution of the conditions of production themselves. The latter distribution, however, is a feature of the mode of production itself. . . . If the material conditions of production are the co-operative property of the workers themselves, then this . . . results in a different distribution of the means of production . . . ." Put another way, the most important thing is the direct control of the means and conditions of production by the producers themselves, rather than by the State or by the Party.

To phrase it in a little romantic way, the first stage of communism is building what we imagine; building the world we want to live in. It will take time to build it; it will be hard, both before and after the revolution. The ‘higher’ stage of communism is the limit of what we can imagine right now, but after the revolution, it will cease to be ‘higher’, because we will imagine another stage which is ‘higher’ than what we imagine right now. When we reach that point, there will be new things we imagined. Communism will always be self-transcendent, we will always move towards our imagination...