The title of Things Fall Apart by Chinua Achebe was taken from a poem by W. B. Yeats called "The Second Coming." The specific lines it references are:
Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world ...
These lines refer to the end of a dispensation in which an old order is falling apart and a new one has yet to arise. This means that the world has no sense of communal ethics or shared beliefs, but instead is in a state of anarchy or disarray.
In one sense, the title refers to the transition to British colonialism and then back again to self rule, with the Ibo being caught between their native religion and Christianity and tribalism and colonialism. In another sense, the protagonist himself is also a man suspended between these two worlds, and because of some of his bad choices, his life also falls apart.