The Council describes this relationship in Gaudium et Spes 76, saying that both Church and State are autonomous and independent of each other, yet both are devoted to nurturing the personal vocation of man, though under different titles.
The totalitarian States of the last century attempted to separate the Church from the State and extinguish it; for them the world belonged exclusively to the State. The Church's view is clearly in conflict with this, because for the Church the world is both a task and a challenge. This is particularly the case for lay people who are called to Christianise society and culture.
Today again, some politicians are trying to redefine the relationship between Church and State, colouring the Church's position as one of purely subjective faith (or opinion). We have seen this for instance from Barack Obama and Tony Blair. And, at the same time we have witnessed a certain passivity from believing citizens and a failure to defend basic human truths and rights. Indeed, we have also witnessed great efforts in society (for instance, the media) to intrude in people's consciences and to lead them to stop believing in Christ.
Here today lies the great challenge for the Church: a new evangelisation, which must begin with the task of rehumanising the world.
(Incidentally, the German wartime theologian Bonhoeffer said that the Christian's purpose in the State is to try to make the State a better State. He was executed by the Nazis two weeks before the end of WWII.)