The following definition of “anarchy/anarchist,” originally published in the 1930s, is taken from Sebastien Faure’s Encylopédie anarchiste. Faure was an advocate of “anarchist synthesis,” which sought to combine the best elements of anarchist communism, anarcho-syndicalism and individualist anarchism. The article on “anarchist synthesis” in the Enclyopédie anarchiste was written by Faure’s collaborator, the Russian anarchist, Voline, and is reprinted in Anarchism: A Documentary History of Libertarian Ideas, Volume One: From Anarchy to Anarchism (300CE-1939).
There is not, and there cannot be, a libertarian Creed or Catechism.
That which exists and constitutes what one might call the anarchist doctrine is a cluster of general principles, fundamental conceptions and practical applications regarding which a consensus has been established among individuals whose thought is inimical to Authority and who struggle, collectively or in isolation, against all disciplines and constraints, whether political, economic, intellectual or moral.
At the same time, there may be – and indeed there are – many varieties of anarchist, yet all have a common characteristic that separates them from the rest of humankind. This uniting point is the negation of the principle of Authority in social organizations and the hatred of all constraints that originate in institutions founded on this principle.
Thus, whoever denies Authority and fights against it is an anarchist.
Robert Graham
Robert Graham's Anarchism Weblog 
No, whoever denies Authority and fights against it is NOT an anarchist.
Republicans fight against Democratic authority.
Protestants fight against Catholic authority.
Marxists fight against Capitalist authority.
Usually those fighting against authority are those who would prefer another authority or their own authority.
Anarchists fight against the concept of authority, and nor do we deny it, we just don’t take it.
The reference is to all authority, not just specific kinds. Republicans, Protestants, Catholics and Marxists do not deny all authority and fight against it. It is not enough to be an anarchist simply to refuse to assume authority over someone else. Many people do that for a variety of reasons, none of which involve a rejection of authority itself. Nor do anarchists simply fight against the concept of authority. They refuse to recognize its legitimacy and oppose those who exercise it under whatever pretext, whether that be the Republic, God, the Church, the Party or even the Revolution.
I would agree with rob on the point that Anarchists refuse the legitimacy of authority. We refuse to accept authority as legitimate lifestyle. However, it must be taken into account that there ARE some forms of legitimate authority. The famous example is that of a parent grabbing a three year old by the arm so that the toddler doesnt walk out into the street and get hit by a car. This type of authority is certainly legitimate, and I find it hard to believe anyone, including anarchist, would have a substantial argument against it. A good example of illegitimate authority is that of the American President. There is nothing that gives authoritarian rights to one man to make choices that affect a mass population against their will (wether those against it are the majority or the minority). (:
The example of a parent restraining a child to prevent him or her from being hit by a car is not a case of someone exercising authority over someone else by issuing a directive which the child believes to be legitimate and therefore obeys. Rather, it is the exercise of power over the child by someone of superior strength. That said, I would expect most anarchists to regard the exercise of such power to be justified in these limited circumstances.