How did immigration alter the nation's population and shape its politics?

It is important to remember who is willing to leave their own country to seek a life in the unknown. Those who inherit land, wealth, and businesses, or have saleable skills, or social position, have little interest in immigrating, since they are reaping the benefits of being higher on the social and economic hierarchy in their home countries. So those who immigrate are those who have little to nothing, and few if any options, and may face homelessness, starvation, and death if they remain in their own countries. This makes them very desperate, as well as willing to work for subsistence level wages. If you own a business, these are your cheapest possible laborers.
Now, what was happening in the United States during the 1800s? The westward expansion had filled and settled the midwestern states and a growing cattle industry existed in the area which became Texas, and most of the rest of the country was still an agrarian economy, but industrialization and the expansion of the railroads and telegraph, in addition to technological advances in agriculture and the industrial revolution, began to cause mass migrations to the cities. The reason new immigrants settled primarily in the cities, is because that is where the jobs and opportunities were. After the civil war, former slaves also migrated north to cities. The new immigrants from Europe and the former slaves were subjected to the miserable conditions of the new industrial cities, and the predation of the industrialists. These are the very populations that Marx and Lenin had in mind when calling for the "Workers of the world to unite and throw off your chains". So the political leanings of the cities were more socialistic than the yeoman farm populations. This divide was also exacerbated by the Protestant vs. Catholic population differences between rural and city populations.
But not all immigrants were welcomed with open arms, leading to anti-immigrant sentiment against targeted populations, but eventually, resulting in The Immigration Act of 1924, which set immigration quotas by country and even banned immigrants from some countries. It is important to remember that around 1900, about one in seven residents in the country were first generation immigrants. This high proportion of new immigrants seems to elicit an anti-immigrant backlash, as we also see today. But it is also important to remember that if immigration enriches you, then you're likely to be "for" it, but if you feel that it harms you economically, then you're likely to be "against" it.
For more reading, see:
Marx, Karl. "The Communist Manifesto"
Sinclair, Upton. "The Jungle"
Weber, Max. "The Protestant Ethic and the Spirit of Capitalism"

There have been multiple waves of immigration to the United States. Historically, the American continent was the last to be inhabited by humans. During the Wisconsin glaciation, ca. 50,000 to 11,000 years ago, a land bridge enabled the first immigrants, who are now called Native Americans, to move across a land bridge from Siberia to Alaska. Thus one can say that the entire population of the American continent is the result of immigration rather than evolution.
Although there may have been some Norse exploration and settlement of North America in the tenth and eleventh centuries, no durable colonies were founded and its effect would have been minimal.
Another major wave of immigration was that of the Athabascan peoples who became the Apache and Navajo from Alaska and Canada into the United States in the fifteenth and sixteenth centuries. Although the Navajos borrowed a more settled culture from the Pueblo dwellers already inhabiting much of the southwest, the Apaches remained nomadic and had a strong role in opposing Spanish colonization. The Navajo culture brought a deep spiritual connection to the land which continues to inspire much of southwestern environmentalism.
Early Spanish conquests gave rise to substantial Hispanic population in Florida, Alabama, Mississippi, and much of the southwest. Early conflicts between English-speaking and Spanish-speaking European immigrants have shaped many of the present attitudes towards immigration and conflicts between Anglo and Hispanic political groups. Another major issue is that early Spanish settlers were Roman Catholic while many of the English settlers in New England were Protestant, setting the stage for Protestant–Catholic conflicts in much of the nineteenth and early twentieth centuries, especially over immigration.
Successive waves of immigration, including forced importation of African American slaves, and voluntary immigration of Asian, European, and South American people have dramatically increased the racial and cultural diversity of the United States and contributed to its economic growth. Despite occasional anti-immigrant backlash periods in which earlier immigrant groups fight to keep out more recent immigrants, in general the United States can take pride in its ethnic diversity created by successive waves of immigration.

During the 1800s, immigration mainly stemmed from Germany and Ireland, particularly after Europe struggled through a series of political and economic shocks and Ireland faced a potato famine in the 1840s. Nineteenth-century immigrants were largely Catholic, and they settled at first in cities. The Irish mainly settled in Northeastern cities, such as Boston and New York, while Germans also settled in these cities and in Midwestern cities such as Milwaukee and St. Louis.
Immigrant groups tended to vote for Democrats because the Democratic Party was the party of the working person, and immigrants tended to work either building railroads, laboring in factories, or building canals, among other jobs. The Republican Party, on the other hand, became the party of the elite and was associated with Protestants, while Catholics (and later, Jews) often voted for Democrats. Some Republicans were nativists—that is, they favored anti-immigration policies—while Democratic political machines such as Tammany Hall in New York courted immigrant votes and often won their votes by doling out favors. In the late 1800s and early 1900s, immigrants arrived in the United States from eastern and southern Europe. These people included Italians, Jews, Greeks, Poles, and others. These groups also often tended to favor urban areas and to vote Democratic.