The Pilgrims famous in American lore for sailing across the ocean in the Mayflower and settling near Plymouth rock in Massachusetts left England primarily in response to religious persecution.
After James I came to the English throne, his high church and Roman Catholic leanings increasingly alienated dissenting groups of Protestants. The Puritan sect William Bradford, first governor of the Plymouth colony, belonged to sought to return to the simplicity of the early Christians and shed the "popish" splendor, ritual, and ostentation that increasingly characterized the Church of England under James.
However, church and state were closely entwined in England. To reject the Church of England was to reject the state and, at least in theory, the authority of the king. This was potentially treason. The state insisted that its citizens pay taxes, called tithes, to the Church of England and worship on Sundays in its churches, in large part to demonstrate their loyalty to the crown.
The Puritans wanted to do neither, as they rejected the Church of England. As such, they were living outside the law and were subjected to arrest, fines, and imprisonment. This caused many hardships for them. At first, they tried to settle in the Netherlands, which practiced religious freedom, but were concerned that their children were being quickly assimilated into Dutch culture. They wished to keep their English identity, so they returned to England but were once again faced with persecution. Finally, they left for America in order to both keep their English cultural distinctiveness and to be free to worship in peace.