What we consider today as the Pilgrims were Separatist Puritans from England. The Puritans were a Christian sect that arose after The Reformation and the Separation of the Catholic and Anglican church. Puritans believed that Christianity had been corrupted by the Catholic church and that when the Anglican church replaced the Pope with a king, they perpetuated idolatry and corruption. Because they were considered radical and were not followers of the Church of England, they were not allowed to publicly practice their beliefs which called for a more literal interpretation of the Bible and a rejection of earthly desires and connections (similar to monks and nuns without earthly possessions and with unadorned, plain clothing). Because they weren't allowed to practice their beliefs, they moved to different locations in attempts to exercise their religious freedom.
Originally, they moved to Holland but life and conditions for them were very difficult for them there. There was also the threat of imminent war in the area.
They then moved to America in the hopes of "propagating and advancing the gospell of the kingdom of Christ in those remote parts of the world" and to separate from what they saw as the corruptible extravagances of Europe. Although they themselves did not have the money to fund their voyage, the Pilgrims organized passage by partnering with the Virginia Company as part of a colonial enterprise.