The exact nature of immigration as an issue in American politics is dependent upon who you talk to. For some, the issue is only illegal immigration and not legal immigration. Some people believe that both are an issue, while others think that neither is a major problem.
There is a division on the left concerning illegal immigration
Although illegal immigration is a decisive issue, groups that oppose each other can be found within both major parties. There are those in the Democratic party that want immigration reform for those they think of as being undocumented workers. These reforms often include a path to citizenship for those who are currently here illegally. Also in the Democratic party are union members who do not like this idea. They see workers who are here illegally as representing an increase in the supply of labor. This, in turn, lowers the demand for labor; therefore, the average wage of the American worker is stagnant or even in decline in certain sectors of the economy.
There is a division on the right concerning illegal immigration
On the right side of the political equation are those who represent large companies and corporations that enjoy lower labor rates. Labor costs are a big part of the expense of doing business. Those here illegally help to keep labor costs down, so profits can stay high. However, in the Republican party, there is also an older, traditional conservative wing of the party. These people see the issue as one of law and order. They want to see many of these workers in the country illegally deported.
There is controversy over legal immigration
This is especially true with the issue of refugees from war torn Islamic countries. Some of these concerns are misplaced, but there are many legitimate concerns over people who are coming here and not being vetted properly. The thinking is that some of the refugees may be our enemies and are only pretending to be refugees, so they can enter the country, and later commit acts of terrorism against us. Right or wrong, this perception has made the issue of legal immigration a hot button issue. Whereas it has mostly been an issue with illegal immigration in the past.
Of course, the opinions above are ones that can be found in the major political parties. There are many other opinions as well. An example of independent party views on immigration can be found in many political books and articles on the subject.