The significance of the title is that from the point where 'An Inspector Calls', the lives of each of the characters in the play have changed forever.
Prior to Inspector Goole's arrival, the Birlings are having dinner and celebrating their great successes to date and the positive view of the future for them and those like them. The family are celebrating the engagement of Sheila Birling, the daughter, to Gerald Croft, son of Lady Croft. Arthur Birling confides in Gerald that he is likely to be knighted himself. We see as an audience that Arthur Birling's perceptions are somewhat askew -he dismsses the idea of war, and extols the virtues of the unsinkable Titanic: both events which had played out before the drama was performed.
Birling had just been explaining to Gerald and Eric that one should be independent and not rely on others.
BIRLING:...But the way some of these cranks talk and write now, you'd think everybody has to look after everybody else, as if we were all mixed up together like bees in a hive - community and all than nonesense.
It is a dramatic foil to the socialist message which the inspector delivers to the family-
INSPECTOR:...We don't live alone. We are members of one body. We are responsible for one another.