Anarchy is a condition of life without the intrusion of governance and the mechanisms and institutions of the state.
Anarchism is the political philosophy which has anarchy as its goal.
The tricky part is that some anarchists believe that the best and only way to achieve anarchy is through anarchism. The implication is that a majority of a given population would necessarily need to be anarchists, and only one type of anarchist. This would result not in anarchy as a condition of existence, but of the institutionalization of a particular type of anarchism. This situation would inevitably breed its own internal logic of conformity and orthodoxy.
By way of contrast, there are many anarchists (those who believe in the desirability of anarchy) who recognize that most non-anarchists, left to their own devices in the absence of the state, would probably choose to live without institutionalized hierarchy -- as humans have done for 99% of our existence. Not to get too metaphysical, but there's probably something to there being a genetic memory along those lines; why else would most insurrections and rebellions that last for more than a couple of days begin to create deliberative and executive assemblies with mandated delegates, rotation of tasks, a (mostly) refusal of professional politicians, etc? Anarchists didn't invent those forms of self-organization, nor do we have an exclusive monopoly on their use. Most reasonable people choose to implement their self-organization along those lines because they understand the justice and solidarity that they entail.
Another way of looking at it is that anarchy does not require a majority (or even a minority) of anarchists in order for it to function (again, there's that 99% of human history to draw on), while the implementation of anarchism absolutely does.
