It depends on the country really and time period.

For example, the Irish migrated due to the potato famine, better job opportunities or maybe even pushed of their farms by British settlers. The Germans were driven by population constraints, lack of farmland throughout Germany, sometimes religious prosecution and again opportunities offered in the US and elsewhere.

Each group had specific reasons for leaving Europe but one common factor seems to be the massive population increase and urbanization of the 19th century. The total population of Europe doubled to over 400 million during the century and this explosion in growth put a great strain on rural areas which could not support such an increase in population. Even with increased agricultural efficiency, the population doubled but the total area of farmland stayed the same.

Many young people were forced to either migrate to cities to look for employment in new modern industries or forced to migrate to the U.S. to either buy farmland (as in the case of the Germans and Scandinavians) or look for new jobs in the cities of the east coast.

That being said European immigration to the U.S. at least, came in a few 'waves' each one was comprised of different proportions of different immigrant groups. Generally, the first batch of immigrants was primarily from the British Isles, Germany and Scandinavia and remained relatively consistent for 50 years or so until the end of the 19th century in which many Italians, Russians, Poles and other Eastern/Southern Europeans came in ever higher proportions and immigration from other countries dwindled off.

This is just a very broad overview of what caused the mass migration from Europe to the U.S. and Argentina and if you have a specific time period or people group in mind I can give a more detailed answer.

Sources: Roger Daniels Coming to America: A History of Immigration and Ethnicity in American Life

Thomas Dublin Immigrant Voices: New Lives in America