European colonization in the Americans and other places in the world (like India or the Far East) brought new cultural ideas and practices that affected the native inhabitants, who then incorporated the new ideas and practices into their own cultures. Let's look at some examples of how this worked.
A prime example of this cultural shift is the Spanish introduction of Christianity to the native populations of Central and South America. As the native people accepted Christianity, they incorporated these new beliefs into their own culture, making their expressions of the Christian faith rather different from those of the Spanish. Through this blend of religion and culture, new cultural patterns emerged.
We can see other examples of new cultural forms in the changed trading practices of North American Native Americans after the arrival of British colonists. The Native Americans began to adopt new values and preferences, including the desires for firearms and alcohol, and new means of communication. Further, as they were pushed off their lands, they often had to change their lifestyles to adapt to their new environment. This included new farming and hunting practices that shifted their cultural practices in different directions.
We can look to the British colonization of India to find non-American examples of cultural change. The British subjected India to a whole new style of governance. The Raj, as the British empire in India was called, established a hierarchy of government that Indians eventually began to enter into through the Indian National Congress. They thus had to adjust their cultural forms to adapt to this new type of government.