In 1860, there were 4.1 million immigrants whom arrived in the U.S. The 4.1 million made up more than 13 percent of the total population: Ireland- 1.6 million, Germany- 1.2 million, and other Western Europe- 900,000 (Scotland and England.) Compare with current stats: 2013 immigrants who make up 28% of the 41.3 million in United States.
Bonus: Andy's grandfather, Thomas Devine Sr., was born in Ireland, and emigrated to the United States in 1852. Andy's mother was Amy Ward, the granddaughter of Commander James H. Ward, the first officer of the United States Navy killed during the Civil War.
Edit - Ireland may have provided the most immigrants during 1860, but the question asked which provided the most immigrants during the five-year period 1860-64.
Hoosier