The major impact that immigration has had over the entire course of US history is to make our society the melting pot that it is today. Of course, all Americans (outside of Native Americans) are immigrants in some way. We have had many major waves of immigration that have brought in large numbers of people from (in rough order that they came) Ireland, Germany, Southern and Eastern Europe, Asia, and Latin America. These people have all tended to assimilate into our society.
In addition, immigrants have helped to push our economy's development. Immigrants provided the work force that drove the Industrial Revolution. They were much of the work force that built the transcontinental railroads. They helped to settle the Great Plains.
Immigration, then, has shaped our society and our economy over the years.