Irreligion is the absence of religion and/or an indifference towards religion. Irreligion sometimes also refers to hostility towards religion, though this is relatively uncommon. Irreligion is distinct from atheism and theism: theists may be religious or irreligious; atheists may also be religious or irreligious. Irreligion is more an attitude towards religion rather than an actual religious position.
Atheism & Irreligion
Atheists in America today are more likely than theists to be irreligious in the sense of simply not having religion. Both atheists and theists are probably equally likely to be irreligious in the sense of being indifferent to religion. Agnosticism, though, may track most closely with irreligion.
In the past, irreligion and atheism have been used synonymously in societies where the only religions of any size were theistic. For example, we read in A Companion to Philosophy of Religion in the section "Early Modern Philosophical Theology in Great Britain":
Besides fideism and rationalism, another response to religious skepticism is, of course, atheism. But although many of the influential expounders of reasonableness endorsed toleration of religious diversity, this toleration did not extend to irreligion. If not unthinkable, atheism was at least unspeakable in Great Britain before the late eighteenth century, subject to both universal social censure and official legal sanction.
Note how "irreligion" and "atheism" are used interchangeably. It was possible to drop the prevailing Christian religion without also becoming an atheist, but from the perspective of orthodox Christians the differences between deism or any other alternatives and outright atheism would have been too inconsequential to pay attention to. Any skepticism of the traditional monotheistic conception of God was as good as — or as bad as — atheism.
Irreligion vs. Social Order
This confluence of irreligion and atheism was not limited to early modern Christian Europe. David Frankfurter writes in his essay "Traditional Cult" in A Companion to the Roman Empire
In Rome, North Africa, Gaul, Syria, Asia Minor, and elsewhere, ‘‘religion’’ revolved around action – gesture, performance, and the fulfillment of ritual responsibilities – whether in the public realm of civic drama or in the privacy of a ‘‘magical’’ rite.
‘‘Irreligion’’ and atheism were under- stood in similar terms: not disbelief or philosophical skepticism but rather the avoidance of ritual responsibilities or even the pursuit of ritual practices that could subvert the social order.
It's the risk of subverting the social order which makes irreligious atheism such a danger. Where the social order is founded on a common set of popular religious beliefs, practices, assumptions, and institutions, participation in that popular religion is what allows a person to be a part of the community.
Choosing to separate oneself from the religious community means separating oneself from the pollical and civil community as well. You may still be physically present, but you can no longer be entirely trusted.
Irreligion & Revolution
On the other hand, when confidence in the social order declines and resistance to the ruling elites becomes a live option, irreligion can become synonymous with that resistance. The irreligious threat to the social order is conscious, deliberate, and revolutionary. Timothy Larsen writes in the Encyclopedia of Nineteenth-Century Thought:
Religion was frequently viewed as aligned with ruling elites and conservative politics, and therefore, conversely, irreligion became a political weapon, a way of furthering radical politics. Radicals, of course, often had good evidence from which to deduce that the Church was one of their main opponents.
In some times and places, it's another religion which provides a measure of ideological organization and justification to resistance to an oppressive social order. Christianity fulfilled such a role before it became the ruling religion in Rome. Protestantism fulfilled such a role during the Reformation.
With the rise of secular philosophy and science, however, irreligion and irreligious ideology could serve the same function. Early on it was just a vague sort of secular, scientific skepticism; in the modern era, it's been secular ideologies like nationalism and socialism.