I generally use "anarchist" to describe my political philosophy. I'm pretty sure I'm using it correctly, but I'm not certain. I haven't had much contact with other "anarchists", just a bit of exposure through history and such.
First off, to me, "anarchism" doesn't mean "no government". Rather it means "no intrinsic authority". What I see among historical anarchists is an opposition to practices that, frankly, aren't all that often practiced any more, in the political realm. I'm referring to rule by bloodline and such, nobility and royalty. I get the impression the early anarchists wanted to do away with royal governance, in favor of a federation of voluntary governments instituted at the local level. Which is to say, they believed in government; they just wanted to do away with imposed external authority.
But I do see our current economic relations as having a great deal of externally imposed authority in it... though going into my beliefs about why, and what could be done about it, would be beyond the scope of this essay.
To me, anarchism means the following:
Favoring no unnecessary relationships of authority.
Where authority is necessary, it should be granted by those over whom the authority is exercised, directly and individually, to the greatest extent practicable. So, for example, if we have an economic system that leaves both employers and employees with the same level of market power (we do not, but if we did), the employer-employee relationship would qualify, since it commences by choice of both parties, and can end by the choice of either party.
Where this is impracticable, the authority in question should always be temporary, with a clearly delineated end. For example, the parent-child relationship is necessarily one of authority, since children lack the faculties to make all the decisions one needs to make. But this relationship should be premised on preparing the child to survive outside this relationship, and have a clear end point (the point of their majority). And I mainly include this but just for the parent-child relationship; I can't think of any others.
All this being said, I know there are those for whom Anarchism means "no government", usually detractors who don't actually understand the philosophy... or so I assume. Do I assume incorrectly? Is my use of the term wildly incorrect? I really don't know.
From my perspective "necessary authority" is a meaningless phrase. Authority is always justified to those who support it, and unjustified to those suffering under it. For example, the authority of a particular country to enforce it's borders is "justified" in order to preserve those borders. The authority of a catholic priest is "necessary" to uphold the values of the church against sin and pollution of the faith. But if you don't believe in those institutions then the justification is very silly! Anarchists just take this one step further and realize there is no single true authorized power structure.
Anarchists can do away with authority and just act directly. If some of us agreed to live a certain way it's us deciding that we want to do that. We don't need to hit others over the head with a magic scroll or manifesto to prove we are justified, and we don't need to ask their permission first. Likewise, if someone wants to push us around we don't give them the benefit of the doubt through authority. They're just another person trying to push us around, whether they're a government agent or a highway bandit.
How you approach others is what makes your world the way it is. If you want to treat others as equals, directly, and engage with them without an intermediary we could say you're an anarchist. If you think you need government (an authority, legitimate justified power) to push others around, i'd say you're not.
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