The people who are now popularly known as the 'Pilgrim Fathers', is a name that became used by the 18th century for the first colonists who sailed on the Mayflower from Plymouth, England to Massachusetts in 1620. They would not have called themselves by that name.
They were the people who set up the Plymouth Colony in present-day Plymouth, Massachusetts, United States.
They were people who held very strict religious views - views that were not welcomed in England at this time. Under the 1559 Act of Uniformity, it was illegal not to attend official Church of England services. As a result, many of these people who wanted to worship apart - they were separatists - moved to Holland. But when they were in Holland, they feared that they would lose their language and chosen way of life, so they decided to seek a new life in a colony of the New World.
You have to keep in mind that the very first English colony in North America had only been set up at Jamestown in Virginia in 1607, so their decision was amazingly brave. The land they were going to was far to the north of Jamestown, it was mostly unknown, and all they had to go on were reports by whalers and fishermen who used to call in to the coast of the New World from time to time. In fact, these fishermen had already caused bad feeling with the American Indians, but the new colonists were not to know that.
The ships they were eventually able to hire for this journey were the Mayflower and the Speedwell. But they were old ships at the end of their working lives. That is why they were relatively cheap to hire, but their age added to the danger.
They first set out on August 15th from Plymouth, but the Speedwell kept developing leaks, possibly made by the crew who were afraid of the journey. Whatever the cause, the Speedwell had to be left behind, but this meant many provisions had to be left behind, too. All the delays in trying to repair the Speedwell meant their rations were going down all the time, making provisions for the journey perilous.
Some people had to be left behind because they would not all fit onto the Mayflower, and the smaller party finally sailed on September 16, 1620.
The ship was not suited to travelling in a part of the world where Westerlies produced a more or less continuous headwind, so the ship had to keep tacking about, making the journey longer. It was not suited to Atlantic storms either.
As a result of all of this the ship finally reached the New World coast in early December.
They sent a small party ashore to look for a good place to set up a colony. What they found were some remains of a European fisherman's house and some mounds which they dug open. These proved to be Indian graves. They also found a village, but the people had run off. Being short of food, they took some of the food from the graves and the village, intending to repay them later. However, when they met with the village people, they shot arrows at the colonists, who fired back using muskets. It was not a good start.
When the native peoples were encountered again, some months later, the colonists discovered that some of them spoke English as a a result of contact with the fishermen from earlier times.
The place they finally chose as a location for their settlement was reached on December 21, 1620 (now called Forefathers' Day in Massachusetts) This was Plymouth Rock. The colonists were not attacked at this place because the local people had mostly died from plague brought over accidentally by earlier European trades and fishermen.
There was snow on the ground, so it was important to get some shelter made. The first house was up by January 19. Each family was allocated a plot one-half rod wide and three rods long for each household member. The families then built their own dwellings. As a result, the settlement was in place with its houses by early February.
But lack of good food, and the inability to treat common diseases meant that many people died. The first house built became a hospital. By the end of February thirty-one of the company were dead. Between the landing and March, only 47 colonists had survived the diseases they contracted on the ship or after. During the worst of this period, only six or seven of the group were able to feed and care for the rest. Half the Mayflower crew also died.
Fortunately for the settlers, the local people were willing to allow them to stay, and they signed a peace treaty on March 22, 1621. The 'Pilgrim Fathers', had made it through the worst of times, and Plymouth was to become the longest continually settled place in the United States.
The use of the term pilgrims was first made by one of their leaders, quoting the Bible. But he was simply talking about their journey, and they did not think of themselves as pilgrims.
It was only at the end of the 18th century that people began to call these first settlers pilgrims. On December 22, 1798, as part of a celebration of Forefathers' Day in Boston, the first settlers were described as "The Pilgrims of Leyden" (Holland). And from then on, the term Pilgrim Fathers became the popular term of choice, as it remains.