If I recall correctly, the Christian Democratic political movement originated in Europe in the 1920s. For much of the 19th Century, and well into the 20th, liberalism and democracy had been seen as essential secular movements, especially in predominantly Catholic countries, and Christians had been suspicious of them, or had tended to avoid them. As liberalism and democracy became dominant in Europe, Christians and Catholics especially tended to distance themselves somewhat from political involvement and public life.
Christian democracy started as a radical reappraisal of that attitude, and sought to explore how Christians could involve themselves in public life, and express Christian values through political engagement, in a way not inconsistent with the principles of liberalism and democracy.
It remained, I think, a fairly philosophical movement with limited political effect until after the Second World War. There was then a widespread feeling that pre-war political and public movements founded on traditional liberal and democratic principles had proved unable to deal effectively with the rise of fascism in Italy or Naziism in Germany; many of them had been positively co-operative. People were looking for a political philosophy which was firmly democratic but at the same time was founded on values which were absolutely inconsistent with fascism (or communism), even if these movements gained power through democratic means. Christian democracy seemed attractive from this point of view.
Although it originated among Catholic thinkers, it was never a denominational or confessional movement. It seeks to promote values which emerge from Christian thinking, but anyone who adheres to those values is welcome to participate in the movement, whether or not he is a Christian, and it seeks popular support for its values not on the grounds that they are Christian but on the grounds that they will serve society well.
In Europe the movement is, unsurprisingly, strongest in those countries which have emerged from fascism. It has never really taken root in the Anglo-Saxon world; centre-right parties in Anglo-Saxon countries may affiliate to the Christian Democratic International, but it is not a particularly important affiliation for them. In some countries the name has been hijacked by fringe parties who are mainly concerned with advocating very conservative policies on sexual morality, but have little interest in broader social or economic issues. They are often explicitly concerned to promote Christianity as such, in a way that mainstream Christian Democracy is not. These parties tend not to be affiliated with the Christian Democratic International, or to buy into the core Christian Democratic values of freedom, responsibility, equality, justice, solidarity and subsidiarity.