Around 30 years before Shakespeare was born, King Henry VIII was denied a divorce from his wife, Catherine of Aragon, by the Pope.  

Henry responded by decreeing that no one, including the Pope, could tell the King what he could and couldn't do; Henry also formed the Church of England and declared himself the head of the new Protestant church.

 
Following King Henry's death in 1547, Queen Mary I (Henry's eldest daughter) brought back Catholicism during her 5 year reign; due to some of her policies she was given the nickname "Bloody Mary."
Queen Elizabeth took the throne when Mary died in 1558.  Elizabeth made the Protestant religion England's official religion, instituting the Book of Common Prayer and making Sunday service mandatory.  During Elizabeth's reign, the penalty for professing atheism or criticizing the Church was death by gallows.

LINKS:

Protestant England (a more extensive look at the rulers and the Reformation)
The Reformation   (about Catholic Church history and Martin Luther)

 
Two main groups formed while the Church of England was being established during Queen Elizabeth's reign--the Puritans and the English Roman Catholics (Papists).  
The Puritans desired to purify the church removing all traditions of Roman Catholic origin.
The Papists desired a return to Roman Catholicism.