TL:DR the state is a tool for the suppression of one class by another. After "lower stage communism", where the bourgeoisie is suppressed, their ideology is erased, and the means of production have been massively developed, we see the state wither away. We cannot predict what this will look like though beyond "transforming labor into 'life's prime want'" and producing enough of what everyone needs that there is no competition of shortage; thus proucing the maxim "from each according to their ability, to each according to their needs". We cannot predict exactly what this will look like because of the limited insight our material conditions offer us. Also read State and Revolution and Critique of the Gotha Program

The state is a special tool for the oppression of one class by another, and as such the mission of the communist is to lead the proletariat into seizing the state from the bourgeoisie and in doing so, beginning their class dictatorship. This is 'lower stage communism', usually called 'socialism'. In this society, the state exists as it does now, though it has undergone revolutionary changes. From State and Revolution:

In the Critique of the Gotha Programme, Marx goes into detail to disprove Lassalle's idea that under socialism the worker will receive the “undiminished” or "full product of his labor". Marx shows that from the whole of the social labor of society there must be deducted a reserve fund, a fund for the expansion of production, a fund for the replacement of the "wear and tear" of machinery, and so on. Then, from the means of consumption must be deducted a fund for administrative expenses, for schools, hospitals, old people's homes, and so on.

Instead of Lassalle's hazy, obscure, general phrase ("the full product of his labor to the worker"), Marx makes a sober estimate of exactly how socialist society will have to manage its affairs. Marx proceeds to make a concrete analysis of the conditions of life of a society in which there will be no capitalism, and says:

"What we have to deal with here [in analyzing the programme of the workers' party] is a communist society, not as it has developed on its own foundations, but, on the contrary, just as it emerges from capitalist society; which is thus in every respect, economically, morally, and intellectually, still stamped with the birthmarks of the old society from whose womb it comes."

It is this communist society, which has just emerged into the light of day out of the womb of capitalism and which is in every respect stamped with the birthmarks of the old society, that Marx terms the “first”, or lower, phase of communist society.

The means of production are no longer the private property of individuals. The means of production belong to the whole of society. Every member of society, performing a certain part of the socially-necessary work, receives a certificate from society to the effect that he has done a certain amount of work. And with this certificate he receives from the public store of consumer goods a corresponding quantity of products. After a deduction is made of the amount of labor which goes to the public fund, every worker, therefore, receives from society as much as he has given to it.

And directly from Critique of the Gotha Program

Here, obviously, the same principle prevails as that which regulates the exchange of commodities, as far as this is exchange of equal values. Content and form are changed, because under the altered circumstances no one can give anything except his labor, and because, on the other hand, nothing can pass to the ownership of individuals, except individual means of consumption. But as far as the distribution of the latter among the individual producers is concerned, the same principle prevails as in the exchange of commodity equivalents: a given amount of labor in one form is exchanged for an equal amount of labor in another form.

So in lower stage communism we see a society not all that foreign to our own, merely with the means of production held in common and without the bourgeoisie siphoning value from the top. Also we will see the vast developing of the means of production once they are held in common because of a lack of the monopoly inherent to the capitalist system. But how do we reach higher stage communism? The stateless, classless, moneyless society?

From Marx:

In a higher phase of communist society, after the enslaving subordination of the individual to the division of labor, and therewith also the antithesis between mental and physical labor, has vanished; after labor has become not only a means of life but life's prime want; after the productive forces have also increased with the all-around development of the individual, and all the springs of co-operative wealth flow more abundantly – only then can the narrow horizon of bourgeois right be crossed in its entirety and society inscribe on its banners: From each according to his ability, to each according to his needs!

Lenin:

The economic basis for the complete withering away of the state is such a high state of development of communism at which the antithesis between mental and physical labor disappears, at which there consequently disappears one of the principal sources of modern social inequality--a source, moreover, which cannot on any account be removed immediately by the mere conversion of the means of production into public property, by the mere expropriation of the capitalists.

This expropriation will make it possible for the productive forces to develop to a tremendous extent. And when we see how incredibly capitalism is already r*tarding this development, when we see how much progress could be achieved on the basis of the level of technique already attained, we are entitled to say with the fullest confidence that the expropriation of the capitalists will inevitably result in an enormous development of the productive forces of human society. But how rapidly this development will proceed, how soon it will reach the point of breaking away from the division of labor, of doing away with the antithesis between mental and physical labor, of transforming labor into "life's prime want"--we do not and cannot know.

That is why we are entitled to speak only of the inevitable withering away of the state, emphasizing the protracted nature of this process and its dependence upon the rapidity of development of the higher phase of communism, and leaving the question of the time required for, or the concrete forms of, the withering away quite open, because there is no material for answering these questions.

The state will be able to wither away completely when society adopts the rule ** "From each according to his ability, to each according to his needs", i.e., when people have become so accustomed to observing the fundamental rules of social intercourse and when their labor has become so productive that they will voluntarily work according to their ability."The narrow horizon of bourgeois law", which compels one to calculate with the heartlessness of a Shylock whether one has not worked half an hour more than anybody else--this narrow horizon will then be left behind. There will then be no need for society, in distributing the products, to regulate the quantity to be received by each; each will take freely "according to his needs".**

And so we have a basic understanding of how higher stage communism will appear and what its basic definition is. Lower stage communism is our goal so we might massively develop the means of production without the interference of the bourgeoisie or the inefficiencies inherent in the capitalist system while also slowly freeing ourselves of bourgeois ideology. Once this is done, we will begin higher stage communism, but we must remain historical materialists, as Lenin writes:

But how rapidly this development will proceed, how soon it will reach the point of breaking away from the division of labor, of doing away with the antithesis between mental and physical labor, of transforming labor into "life's prime want"--we do not and cannot know.

That is why we are entitled to speak only of the inevitable withering away of the state, emphasizing the protracted nature of this process and its dependence upon the rapidity of development of the higher phase of communism, and leaving the question of the time required for, or the concrete forms of, the withering away quite open, because there is no material for answering these questions.

We cannot predict higher stage communism anymore than the Romans could have predicted modern capitalism because our material conditions and our thought based in this reality is limited. We can only lay the foundations for the withering of the state.