When communists say we want to abolish the state, we do not mean we want to cease all its functions. We want to abolish it as an instrument of class rule, as an entity that is above and alienated from the rest of society. The socially useful tasks of current states, whatever they may be, can be integrated into civil society.

My understanding is that in Marxism the state has a specific definition: it is the special apparatus by which one class enforces its will and oppresses another. Currently it is used by the bourgeoisie to oppress the working class. After a working class revolution, another type of state develops, one which is used to oppress the bourgeoisie. But this state is transient: after the dissolution of the bourgeoisie, then (in principle) class ceases to exist, and there is no need to have any sort of apparatus by which one class oppresses another, so almost by definition the state ceases to exist. But the lack of a state in a Marxist sense does not preclude the existence of some form of centralized administrative body which allocates resources and organizes the economy and production.

You abolish the state, not the government.

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.

Insofar as decisions about society need to be made, the people as a whole have the right to make them.

In openly undemocratic states, this right is denied outright.

In liberal democracies, this right is, in theory, surrendered to the state as part of a social contract. In practice, it is denied outright.

Without a state, it is still possible to have a form of organization that does any of the important functions usually fulfilled by states, but that organization is unlike a state in that it does not hold any power independent of the will of the people, not even by asking them to surrender it consensually.

From ABC of Communism by Bukharin, a sort of communist party primer:

Distribution in the communist system:

The communist method of production presupposes in addition that production is not for the market, but for use. Under com munism, it is no longer the individual manufacturer or the individual peasant who produces; the work of production is effected by the gigantic cooperative as a whole. In consequence of this change, we no longer have commodities, but only products. These products are not exchanged one for another; they are neither bought nor sold. They are simply stored in the com munal warehouses, and are subsequently delivered to those who need them. In such conditions, money will no longer be re quired. 'How can that be?' some of you will ask. 'In that case one person will get too much and another too little. What sense is there in such a method of distribution?' The answer is as follows. At first, doubtless, and perhaps for twenty or thirty years, it will be necessary to have various regulations. Maybe certain products will only be supplied to those persons who have a special entry in their work-book or on their work-card. Subsequently, when communist society has been consolidated and fully developed, no such regulations will be needed. There will be an ample quantity of all products, our present wounds will long since have been healed, and everyone will be able to get just as much as he needs. 'But will not people find it to their interest to take more than they need?' Certainly not. Today, for example, no one thinks it worth while when he wants one seat in a tram, to take three tickets and keep two places empty. It will be just the same in the case of all products. A person will take from the communal storehouse precisely as much as he needs, no more. No one will have any interest in taking more than he wants in order to sell the surplus to others, since all these others can satisfy their needs whenever they please. Money will then have no value. Our meaning is that at the outset, in the first days of communist society, products will probably be distributed in accordance with the amount of work done by the applicant; at a later stage, however, they will simply be supplied according to the needs of the comrades. It has often been contended that in the future society everyone will have the right to the full product of his labour. 'What you have made by your labour, that you will receive.' This is false. It would never be possible to realize it fully. Why not? For this reason, that if everyone were to receive the full product of his labour, there would never be any possibility of developing, expanding, and improving production. Part of the work done must always be devoted to the development and improvement of production. If we had to consume and to use up everything we have produced, then we could never produce machines, for these cannot be eaten or worn. But it is obvious that the bettering of life will go hand in hand with the extension and improvement of machinery. It is plain that more and more machines must continually be produced. Now this implies that part of the labour which has been incorporated in the machines will not be returned to the person who has done the work. It implies that no one can ever receive the full product of his labour. But nothing of the kind is necessary. With the aid of good machinery, production will be so arranged that all needs will be satisfied. To sum up, at the outset products will be distributed in proportion to the work done (which does not mean that the worker will receive 'the full product of his labour'); subsequently, products will be distributed according to need, for there will be an abundance of everything.

Administration in the communist system:

In a communist society there will be no classes. But if there will be no classes, this implies that in communist society there will likewise be no State. We have previously seen that the State is a class organization of the rulers. The State is always directed by one class against the other. A bourgeois State is directed against the proletariat, whereas a proletarian State is directed against the bourgeoisie. In the communist social order there are neither landlords, nor capitalists, nor wage workers; there are simply people - comrades. If there are no classes, then there is no class war, and there are no class organizations. Consequently the State has ceased to exist. Since there is no class war, the State has become superfluous. There is no one to be held in restraint, and there is no one to impose restraint. But how, they will ask us, can this vast organization be set in motion without any administration? Who is going to work out the plans for social production? Who will distribute labour power? Who is going to keep account of social income and expenditure? In a word, who is going to supervise the whole affair? It is not difficult to answer these questions. The main direction will be entrusted to various kinds of book-keeping offices or statistical bureaux. There, from day to day, account will be kept of production and all its needs; there also it will be decided whither workers must be sent, whence they must be taken, and how much work there is to be done. And inasmuch as, from childhood onwards, all will have been accustomed to social labour, and since all will understand that this work is necessary and that life goes easier when everything is done according to a prearranged plan and when the social order is like a well-oiled machine, all will work in accordance with the indications of these statistical bureaux. There will be no need for special ministers of State, for police and prisons, for laws and decrees - nothing of the sort. Just as in an orchestra all the performers watch the conductor's baton and act accordingly, so here all will consult the statistical reports and will direct their work accordingly. The State, therefore, has ceased to exist. There are no groups and there is no class standing above all other classes. Moreover, in these statistical bureaux one person will work today, another tomorrow. The bureaucracy, the permanent officialdom, will disappear. The State will die out. Manifestly this will only happen in the fully developed and strongly established communist system, after the complete and definitive victory of the proletariat; nor will it follow immediately upon that victory. For a long time yet, the working class will have to fight against, all its enemies, and in especial against the relics of the past, such as sloth, slackness, criminality, pride. All these will have to be stamped out. Two or three generations of persons will have to grow up under the new conditions before the need will pass for laws and punishments and for the use of repressive measures by the workers' State. Not until then will all the vestiges of the capitalist past disappear. Though in the intervening period the existence of the workers' State is indispensable, subsequently, in the fully developed communist system, when the vestiges of capitalism are extinct, the proletarian State authority will also pass away. The proletariat itself will become mingled with all the other strata of the population, for everyone will by degrees come to participate in the common labour. Within a few decades there will be quite a new world, with new people and new customs.

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