The term "white people" or simply "white" is a general term in the same way that "black" or "Asian" are non-specific terms regarding cultural identity. During the early years of the United States up until the mid-twentieth-century, some Anglo Protestant white Americans believed that non-Anglo, European-descent American citizens and immigrants were sub-types of the white race.
For instance, immigrants of Irish, Polish and Italian-descent were "secondary whites" compared to the Anglo-Americans. Although considered racist in today's society, this hierarchical and categorical way of thinking regarding the whites of America illustrated the ethnic differences of the white race. Additionally, religion was a factor in one's "whiteness." Such religious intolerance was a form of discrimination or intra-Christian rivalry that is rooted in the atrocities of Catholics and Protestants towards each other in Europe.
The predominantly Catholic Italian, Polish, Greek, Irish, French, and, to a lesser-extent, German immigrants who came to occupy the eastern coast of the United States between the 1800s and 1900s felt discrimination from the Anglo Protestant multi-generational Americans.
Today, there is a generalization of white people that is accepted in American society. "White people" became a term to identify anyone of pure European descent, regardless of their specific ethnic ancestry. In contemporary America, someone who has Italian or even Eastern European Jew ancestry is considered a white person.
So how did the xenophobic hierarchies within the white sub-groups (e.g., English, Irish, German, Italian, Russian, etc.) evolve into a unified white people? Sociologists believe that Jewish and other non-Anglo Protestant whites began to gain social acceptance during the mid-twentieth century. Non-Anglo peoples held corporate jobs, gained social and economic power, as well as produced prominent mainstream celebrities, such as Robert De Niro and Al Pacino.
As white people as a whole began to accumulate more wealth, power, and influence in contemporary American society, the racial hierarchy began to lean towards the discrimination of non-whites, especially African Americans during much of the twentieth- century, and Hispanics and Muslim-Americans today.
It appears that each wave of immigrants across different eras were placed at the bottom of the hierarchy in America. As they rose to the top of that hierarchy, the more they became accepted into the "majority" represented by the whites, who are still the predominant race in America.