The state can only be understood concretly (that is, as a whole) with dialectics. It is a unity of opposites that is moved by the motor that is its main contradiction, the contradiction between the opposites within the whole.
The state has, at its core, two opposites which are economic classes, one of these classes dominates the other. Why? Because it is the economic mode of production from which all forms of social organization spring up, and in the capitalist, servile, slave modes of production, there are two classes, more or less broad, more or less divisible along secondary but important differences. One of these classes produces, exerts its labor-power, which is merely a specific form of the congealed whole of human labor, and the product of its production is appropriated by the other class. The conditions and stipulations of this relationship is what defines a mode of production. (see: Marx)
The state, along with all other forms of social organization, are conditioned from their gestation by these underlying conditions and particularities of the mode of production. The two classes that are formed from the mode of production are the ones that are at the core of the state, and their relationship is a primarily antagonistic one. One class leads, the other follows. One class sows, the other reaps. One class holds political power, the other does not. These opposites are united, associated, and the state cannot exist without this antagonistic relationship because the state is the summation of the implicitly and explicitly coercive tools with which the class order is maintained.
At the same time, the existence of two antagonistic classes in a state, now speaking to our time, the bourgeoisie and the proletariat, implies two types of state. A bourgeois state, and a proletarian state. Keep in mind that the adjective preceding "state" denotes the class that dominates the state machinery, not that there is only one class in that state. This also is a unity of opposites, in which the whole is the global collection of states, rather than a single one. From these two views, it is patently clear that the defining characteristic of a state is the class which dominates it.
The contradiction between the two classes in a state is the motor within it, its movement, its change, is driven primarily by the conflict between these classes. The history of social organization is the history of the class conflict that moves it. The movement of a state can be in many directions, twists, speeds, trends, while its fundamental character remains. This is because the class character of a state is a qualtitative measure, while questions like class consciousness and mobilization, the development of production, the influence classes or sub-classes lose and gain, are quantitative measures.
The movement generated by the contradiction between these antagonistic classes historically points to the ascension of a previously subjugated class to holding economic and political power. But history is not a smooth evolution, it isn't a gradual continuous change from the earliest form of social organization to imperialist capitalism. There are jumps, revolutions, disrupture of continuity. In short, history cannot be understood without qualitative changes, the "big events" that liberal curriculums list like a broadly unconnected and/or random series of events. But these qualitative changes are preceded and followed by quantitative changes which accumulate, build up the necessary tension and conditions for these changes in quality to occur. The Russian Revolution did not happen one day, it was preceded by centuries of class conflict, political radicalization, and other failed attempts at qualitative changes.
This is a why a revolution is a necessity for the qualitative change of the class character of the state, the revolution is the qualitative change which is built up by sometimes centuries of continuous, quantitative change. Reforms, elections, these are quantitative changes which are by themselves unable to effect a qualitative change, but they build up to the conditions and to the actions which can. There is no instance in history in which there has been a disruption in the continuity of the development of a specific form of social organization solely through quantitative, continuous changes.
The proletarian and bourgeois classes are foundationally tied to their positions in capitalist production. The proletariat is the class that produces and has the fruits of that production appropriated, the bourgeoisie is the class that appropriates the fruits of a production foreign to itself. Their positions in a capitalist state is derived from it, the bourgeoisie is that class that holds political, state power, the proletariat is that class which is subjugated. In the moment of a qualitative change in the class character of the capitalist state, when the bourgeois state is replaced as a whole by a proletarian state, the working class that has necessarily lead that jump in quality has negated itself. It has turned the class relations that defined it on their head. It is the proletariat the class that produces and takes full control of the fruits of its production, and consequently, it will also be the class that hold political power, that controls the state.
But the bourgeois class doesn't vanish into thin air, it still exists, subjugated. The qualitative change that has happened is the change in the class character of the state, not in the disappearance of a class. The proletarian state is still a unity of opposites, the same opposites that were associated in the bourgeois state, but reversed in their relationships. The series of quantitative changes that leads to the disappearance of the bourgeois class still has to happen, now energized by the contradiction between the dominant proletariat class and the subjugated bourgeois class. And there still needs to be the series of changes driven by the contradictory relationship between the conjunction of proletarian states, and the conjunction of bourgeois states. This period is socialism, a transition, one that cannot keep still, much like any other period in the history of class antagonisms is unable to keep still. And just like these, its movement is neither straight nor constant.
The qualitative change that is the abolition of the bourgeois class concretly will have two simultaneous consequences. The proletariat, being defined by its antagonistic relationship to the bourgeoisie will also necessarily disappear as a class. It negated itself in the moment of its revolution, and it continued to do so in the process that moves towards the abolition of the bourgeois class, until it superseded itself, through that negation, and disappeared. At the same time, the state, which has as its core characteristic the unity of opposites, class society, will also necessarily disappear, since a state cannot exist without classes to wield it and classes to subjugate.
This is, as I understand it, the materialist and dialectical conception of the state, through its main components: The unity of opposites, movement through contradiction, transformation of the quantitative into the qualitative and viceversa, and the abolishment of the old through its negation, a negation which "does not merely throw to one side, [which] abolishes by first overcoming"