It was not all Europeans; much of the immigration was male-based, similar to current Latin American immigration patterns. In fact, many of the Europeans that came to America eventually returned back to their homelands and some people made repeated visits to America. Up to 1/3rd of European arrivals in America returned to their homelands during this time period.
I think there were a variety of reasons for immigration. My theory is that America, having a head start in industrialization, was attractive to people that lived in rural Europe that had enough capital to immigrate but not enough to enter the upper-class. Another thing was immigration was rather unlimited from Europe.
It also didn't help that much of Europe had incompetent governments lacking the social services we take for granted. Three countries that stand out for me are Austria-Hungary, the Russian Empire, and Italy. These 3 countries were multi-lingual, agrarian , politically divided, and politically unstable.
Germany had a huge diaspora but emigration ended around 1880s. From what I read, German emigration declined when Germany implemented social services we are familiar with like social security and universal healthcare, I think it's fair to say that these social services make emigration unattractive.