Although caesaropapism is an idea that is much more connected to the Byzantine Empire than to Europe, it has had a great impact on European society. This is largely because various European rulers have tried to implement that idea, leading to major conflicts between church and state.
Caesaropapism is the idea that the secular authority in a country should also have control over the religious establishment of that country. This idea is most commonly connected to the Byzantine Empire, where emperors had the right to do things like appointing Patriarchs. However, there were efforts to make this happen in the West as well. For example, there was a time before the Reformation when European rulers tried to exert control over the papacy. The main example of this was when the French kings had a great deal of control over the papacy when the popes were living at Avignon. Perhaps the greatest example of caesaropapism in Europe came in England. There, King Henry VIII launched the English Reformation largely so that he could institute caesaropapism. The monarchs of England became the heads of the Church of England after Henry.
More generally, there was always a tension in Western Europe between religious and secular authority. Kings would want to take semi-papal powers such as the power to appoint bishops within their countries. The church would resist these attempts. The conflict between church and state was very significant throughout much of European history.