The Anglican church was founded when the English King Henry VIII broke off from the Catholic Church when they would not endorse his divorce from his wife on the grounds that she was not giving birth to male errors but of females - the Catholic Church said he had no true justification for a divorce so they put their foot down. He responded by essentially saying, "Fine, I will make my own church and they will approve of my divorce". The churches are very similar in doctrines and church hierarchy, they both have bishops and such, but the Catholics are headed by the Pope in the Vatican while the Anglican church is headed by the Archbishop of Canterbury (where Augustine first spread Christianity in England) but technically the Queen is the leader of the church.
It has been summed up by saying that the Catholic Church is a continental church, all of Europe was led by it, while the Anglican was a nationalized church, where only England recognized it.
England was very unstable in terms of religion. They fluctuated from Anglican to Catholic again under Mary, then Elizabeth came and re-established the church. Cromwell also unsettled things.
A lot of people think people moved to America to escape the Catholic Church, but this is not entirely true. Jamestown was founded mostly by Anglicans who were not persecuted. But the Pilgrims and Puritans who moved to Massachusetts were trying to leave the grasp of the Anglican church. The Pilgrim's were separatists, they believe the Anglican church was totally wrong and irrepairable, so they first moved to Holland where religious freedom was heralded. But Spain and Holland were on the brink of war, and they had other hardships in Holland, so they decided to move to America. The two choices for them were either Virginia or South America. They liked Virginia because the British were nearby and could give them help, but they also feared this because the Anglican Brisih could oppress them in America just like they did in England. Then they thought South America would be good because of the weather, but then Spain would be even closer to them, so they decided it would be better to be close to England then Spain. They didn't land exactly where they wanted to though, they landed north of Virginia, but winter was coming and they needed to set up a settlement so they built up Plymouth.
William Bradford, the famous governor of Plymouth, wrote that the name Pilgrim was a reference to the eleventh chapter of Hebrews.
"All these people were still living by faith when they died. They did not receive the things promised; they only saw them and welcomed them from a distance. And they admitted that they were aliens and strangers on earth. People who say such things show that they are looking for a country of their own. If they had been thinking of the country they had left, they would have had the opportunity to return. Instead, they were longing for a better country - a heavenly one. Therefore God is not ashamed to be called their God, for he has prepared a city for them." Hebrews 11:13
So those men wanted to be free from the Anglican church to set up God's true church. They did not have bishops and cardinals or popes, they were instead what is called Congregational. There are two main types of church structure - Congregational and Episcopal. Episcopal churches have a hierarchy of bishops, priests, popes, etc. These churches include the Anglican church and the Catholic Church. Congregational churches however are usually just a congregation and a minister they chose. So many Congregational churches are non-denominational.
The other religion escaping persecution was the Puritans, but they were not treated quite as harshly as the Pilgrims. The Puritans wanted to "purify" the Anglican church, hence their name, and the main thing they had a problem with was the Book of Common Prayer which was recited in Anglican churches. The Puritans wanted the ministers to be educated and be able to give sermons on topics rather then to just read from a Book of Common Prayer. They did not really have a problem with a church being mixed with the political government, nor did they really have a problem with bishops and such, but once they were in America they became more Congregational and eventually merged with the Pilgrims.
There was only one Catholic that signed the Declaration of Indepdence - Charles Carrol. Many of the founders were Anglican by tradition. One was a trained clergyman from Scotland, but I think he was Presbyterian (Calvinism essentially). George Washington was an Episcopalian. The Episcopal church is the American name of the Anglican church. After the Revolutionary War and America broke off from England, it made sense to untie the bonds with the Anglican church. Instead of evicting the Anglican church, they just changed its name to Episcopal. In fact the very first thing they did after George Washington made his oath when elected was they all marched down the street to the new Episcopal church. Episcopalianism was not a very energetic passionate church, its preachers were not giving dramatic sermons. The Great Awakening happened just a few decades before and led to dramatic sermons, but the Anglican church was largely not affected by this religious revival. The Anglican church remained rather cool and distant and so Congregational churches gained a huge following while the Anglican church lost influence.