Here is a general definition of the word pilgrim: an individual who travels to a holy place for the purposes of practicing religion or expressing devotion to a faith.
In terms of American history, Pilgrims, with a capitalized P, were a group of Puritans who separated from the larger group of Puritans in England. Pilgrims chose to separate into smaller congregations for religious reasons.
These Pilgrims traveled to Plymouth, Massachusetts in 1620 in order to escape religious persecution in England, and they called their settlement the Plymouth Colony. Members of this separatist group ended up in colonial America because they did not approve of the reforms taking place in the Church of England, and they wanted to practice their religion their own way. Pilgrims led the way for other religious separatists in America who desired freedom from societal pressures.