The title Things Fall Apart comes from the poem "The Second Coming" by William Butler Yeats. In this poem, Yeats describes the end of the Christian era which, according to Biblical scripture would result in the second coming of Christ. At the time of this second coming the world will be, at the end of an era, in a tumultuous state and poised for dramatic change. This idea of historical and cultural change (in Yeats, the end of the Christian era and rebirth of a new era) is comparable to the dramatic changes in Okonkwo's life and in Umofian culture in Things Fall Apart.
With the arrival of the missionaries, parts of Umofian (and Ibo) culture are lost and dramatic changes are occurring. In his poem, Yeats also describes a historical moment of dramatic change in which things are out of control and requiring some kind of salvation. Achebe got his title from Yeats' poem because of the similarities: two cultures in disarray and poised for dramatic change. Hence, Achebe also included four lines from Yeats' poem prior to chapter one:
Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the center cannot hold;
Mere anarchy is loosed upon the world.
Yeats' poem was published in 1920, just after the first World War. So, it was written after one of the most violent and far reaching wars in history and at a period of modernization. Things Fall Apart is set during a time when the Ibo/Umofian culture is being dramatically altered. Both the poem and the novel describe such dramatic, historical moments: chaotic changes but perhaps with at least some hope that some change (and/or a "rebirth) will be beneficial.