Basically, the Puritans wanted to "purify" the Church of England of Catholic practices and beliefs.
Wikipedia says that Puritans' belief "makes sense only as certain parts of Reformed theology, i.e. the legacy in theological terms of Calvinism, as it was expounded by Puritan preachers (often known as lecturers), and applied in the lives of Puritans."
So I would look up the differences between Calvinist and Catholicism.
They were pure Calvinists. They believed in 1. the Total depravity of man - all have sinned and all human beings will continue to sin throughout their lives and all things that we do is all tinged with sin. 2. Unconditional grace offered to the elect (the chosen few). That is, no matter how many times the elect sin, or what they do, they will be a member of the elect. 3. Limited atonement- that is, only some people (the elect) will receive the benefits of Christ's atonement for sin on the cross, the rest of humanity would perish. 4. Irresistible grace - the elect can never lose their chosen status, again, no matter how much they sin. 5. Perseverance of the Saints - you cannot ever lose your election.
Most Protestants don't agree with the idea that you cannot lose salvation or that there is nothing we can do to earn it.
Most Catholics believe that St. Peter and his fellow priests are the ones who convey salvation through the sacraments, and that you have to pay for your sins through confession and atonement, something not necessary in Puritanical theology.
Incidentally, the Puritans were not all that pure. They mostly lived in one-room houses, cussed and drank and regularly had sex in front of their children. (How else to have a second one?)
The Puritans had this idea that they were the perfect model for all Christians in mankind. The Puritans basically pictured God as a tyrant, a God who was always angry and upset at his people; the Puritans thought they would constantly be punished by God if they weren't perfect.
Puritans also had this idea that only the predestined could go to heaven( only certain people were good enough in God's eyes) or Calvinism.