When the group of English colonists commonly known as Pilgrims crossed the Atlantic, they brought domestic turkeys with them. When they reached what is now Massachusetts, they found wild birds of the same species living there. At the festival of the first Thanksgiving, they may well have eaten either, or both, wild and domestic turkeys.
The turkey is a native of North America only. Given this, how could European settlers have brought turkeys from Europe to North America? The answer is that turkeys had already been brought from North America to Europe in the sixteenth century. The Pilgrims were already used to eating a distinctly "American" food long before they left England or the Netherlands to come to Massachusetts.
There are several different subspecies or varieties of wild turkey in North America. The southernmost of these subspecies lives in central Mexico. It was the only variety that was domesticated. When the Spanish overran the great Mexica (Aztec) empire in Mexico, they learned about the local foods. Apparently they liked turkeys enough to bring them back to Spain not long after the conquest of Mexico. From Spain, they spread through many other parts of Europe by the end of the sixteenth century, including England.
When the bird reached England in the late sixteenth century, there was a problem with names. The Spanish referred to the birds by a version of their native Mexican name. Hardly any of the English, on the other hand, knew exactly where these birds came from. At that time, the Ottoman or Turkish Empire controlled much of eastern Europe, the Middle East, and North Africa, and was seen as strange and exotic by most people in western Europe. People tended to assume that most strange things, including strange animals and plants, came from Turkey. The large, meaty, but odd new bird became known as the "Turkey bird", because so many people assumed that it came from the empire of the Turks.
The spread of the domestic turkey from Mexico to Europe and then back across the Atlantic to more northern regions of North America is just one small example of a massive movement of domesticated and non-domesticated plants and animals known as the "Columbian exchange". It gets its name from the fact that it was the voyages of Christopher Columbus that started the exchange by encouraging further exploration, conquest, and colonization of new lands by Europeans.
Among other things, Europeans brought wheat, grapes, barley, oats, apples, peaches, oranges, pears, peas, rice, coffee, tea, sugarcane, pigs, sheep, cows, horses, honeybees, and a bunch of diseases from the "Old World"continents of Europe, Africa, and Asia, to the lands they considered a "New World". In return, they took, among other things, corn/maize, potatoes, chili peppers, beans, squash, sweet potatoes, cacao/chocolate, tomatoes, and pineapples back to not only Europe but to other parts of the "Old World". (Turkeys seem to have been the only domesticated animal taken in significant numbers from either American continent to the "Old World".
By doing this, they drastically changed the world's economy. Potatoes in northern and western Europe produced larger quantities of food on the same amount of land than traditional crops like wheat. This eventually led to a great increase of population in northern and western Europe, which probably helped make the industrial revolution possible creating a large number of former peasants who could not find work in rural areas and so came to early factories looking for work.
Corn/maize had little impact in northern or western Europe, but it became a major crop in parts of southern and eastern Europe. Its greatest impact, though, was probably in tropical Africa. Corn varieties that came from wet tropical regions of North and South America grew well in wet tropical areas of Africa, and as the potatoes did in Europe, they produced more food on the same amount of land than traditional local crops like yams. As in Europe, the new crops helped support a major increase in population, but instead of the Industrial Revolution, the extra population in Africa had a much worse fate. All of those extra people probably helped encourage various kingdoms and tribes in western and central Africa to go to war more often, and capture people from enemy kingdoms and tribes as slaves. Many of these slaves were sold to European slave traders, who took them to plantations in the American continents and the Caribbean.
So, just two crops going from the Americas to the "Old World" probably helped create both the African Slave Trade and the Industrial Revolution. The effects of the "Columbian Exchange" certainly went well beyond what was eaten at the first Thanksgiving!