Modern Spelling of Pilgrims Colony is Plymouth Colony.
Pilgrims or the Pilgrims Fathers were religious people settled in Netherlands who were concerned that they might lose their cultural and religious identity if they remained in the Netherlands. So with the help of investors, they arranged a colony in America. The colony was established in 1620 and became the second successful English settlement after Jamestown, Virginia.
So this colony was then named as Pilgrims Colony which after so many years of changes in cultures and societies became Plymouth Colony.