What is an anarchist? I suppose there are probably as many answers to that question as there are anarchists. No bad thing, when you consider that the individual is at the heart of anarchism. However, there is common ground among most anarchists, but explaining that to someone who is not an anarchist can be difficult.
There is a small text by Émile Armand (pseudonym of Ernest-Lucien Juin Armand); 26 March 1872 – 19 February 1963, called Mini-Manual of Individualist Anarchism, though I don't agree with its entirety, what anarchist would agree with the entirety of another anarchist's work, There is a particular paragraph which quite explicitly lays out what is an anarchist.
The anarchist has for enemy the State and all its institutions which tend to maintain or to perpetuate its stranglehold on the individual. There is no possibility of conciliation between the anarchist and any form whatever of society resting on authority, whether it emanates from an autocrat, from an aristocracy, or from a democracy. No common ground between the anarchist and any environment regulated by the decisions of a majority or the wishes of an elite. The anarchist combats for the same reason the teaching furnished by the State and that dispensed by the Church. He is the adversary of monopolies and of privileges, whether they are of the intellectual, moral or economic order. In a word, he is the irreconcilable antagonist of every regime, of every social system, of every state of things that implies the domination of man or the environment over the individual and the exploitation of the individual by another or by the group.The work of the anarchist is above all a work of critique. The anarchist goes, sowing revolt against that which oppresses, obstructs, opposes itself to the free expansion of the individual being. He agrees first to rid brains of preconceived ideas, to put at liberty temperaments enchained by fear, to give rise to mindsets free from popular opinion and social conventions; it is thus that the anarchist will push all comers to make route with him to rebel practically against the determinism of the social environment, to affirm themselves individually, to sculpt his internal statue, to render themselves, as much as possible, independent of the moral, intellectual and economic environment. He will urge the ignorant to instruct himself, the nonchalant to react, the feeble to become strong, the bent to straighten. He will push the poorly endowed and less apt to pull from themselves all the resources possible and not to rely on others.