What most people have in mind when they use the word "anarchist" is the image of a bomb-throwing revolutionist. They do not know about the existence of philosophical anarchists.
The philosophical anarchist is one who mistakenly believes that human beings can live together peacefully and harmoniously without government. He denies the necessity and indispensability of government for the existence of society.
The following example will show why this is an error. Let us suppose three scientists are going off to explore the jungles of the Amazon river. Before they depart, they must agree on some principles that they unanimously accept as governing their decisions while they remain together. Either they must appoint one of them as their leader and agree to abide by his or her decisions, or they must agree on majority rule and abide by the decisions of any two of them against the third. Unless they unanimously accept one or the other of these two principles governing their making of decisions, they will not get very far before they become disunited and at odds. The mission that they are undertaking will fail.
Marxist thought includes the vision of a time when the state and coercive government will wither away and society will endure thereafter peacefully without the existence of coercive government. The dictatorship of the proletariat was thought to be only the penultimate stage in the communist revolution. The ultimate stage was envisioned as an anarchist society -- one from which the evils of government and coercive force had been totally eliminated.
Anarchy exists in the world today only with respect to sovereign states A sovereign prince or state is one that admits no governing superior. Two or more sovereigns in their relation to one another do not constitute a harmonious or peaceful society. In fact, sovereigns are always in a state of war, even when they are not engaged in actual warfare. As Thomas Hobbes correctly observes, war consists not only in actual battle, but in a tendency thereto among sovereigns. We have came to call that state of war, which does not involve actual warfare, as a "cold war." (See War and Peace.)
The League of Nations after the end of World War I in this century, and the United Nations after World War II were steps in the direction of peace, but they did not go far enough. Only if a world federal government were eventually established in the future, a government that would possess a monopoly of coercive force, would we have a state of global peace from which the anarchy of sovereign states and their state of war with one another would be abolished.