Things really got started with Martin Luther, who nailed his 95 theses to a church door (or sent them in a much less dramatic fashion) in 1517. Several theological issues were noted, but the big one was indulgences, which were things you would buy to get your relatives out of purgatory which were used as church funding (Purgatory could be explained in its own right, but basically the church said something like, your friends and relatives will suffer before going to heaven if you don't pay, and then used that money to build cathedrals or whatever), and the idea of faith alone or sola fide, where only faith was needed to go to heaven, as opposed to the Catholic doctrine of faith and good works (charity, basically). Luther thought this was pretty unfair, and his theses were spread by the printing press across Germany and then Europe. Luther was excommunicated in January 1521, but was eventually summoned to the Diet of Worms and declared a heretic by the Holy Roman Emperor, Charles V. Luther was protected by the Elector of Saxony and continued to write. From this point, Protestantism had truly begun, and especially in places like Scandinavia and Germany began to spread. That brings us to the case of the Church of England.
England would cut itself off from the papacy in 1534 and declare King Henry VIII the head of the church in England, but it would keep some Catholic traditions (Henry also used this to get a cheeky divorce or two). After Henry, the Anglican Church began to separate itself from the papacy further, considered Catholic in some ways and Protestant in others.