As part of his somewhat esoteric belief system, Yeats believed that historical change could be conceived in terms of gyres. These are historical epochs, lasting for roughly 2,000 years or so. To help his readers understand this highly complicated concept, Yeats visualizes gyres as interpenetrating cones, with their interpenetration representing the passing of one historical epoch into another.
In “The Second Coming,” one gyre, or epoch of history, is about to come to an end, giving way to another. This is the epoch that has lasted since the birth of Christ. In its place will arise a completely different era, one characterized by bloodshed, chaos, and violence. Yeats senses that this gyre is in the offing by the fact that the falcon cannot hear the falconer. In other words, ordinary people—the falcon—no longer pay heed to the wisdom of their alleged social betters—the falconer—whom Yeats regards as born to rule.
Yeats's theory of the gyre, as set out in “The Second Coming,” is generally thought of as being an anticipation of the rise of fascism, in which all the old certainties of the Judeo-Christian tradition would come under a full-frontal assault. Particular lines, such as “the blood-dimmed tide is loosed,” certainly appear to lend credence to this interpretation.
In any case, it's clear that “turning and turning in the widening gyre” is leading inexorably to a period of human history in which very few of us would want to live.