1. They came to escape poverty.
This was probably the biggest reason. Famine, particularly the Great Potato Famine--an Gorta Mór--of Ireland in the 1840s to early 1850s, compelled people to seek their living in another place.
2. They came for religious freedom.
We've all heard that this is why the pilgrims came to America. Many of our ancestors' narratives passed down contain this reason. But don't forget that in centuries past the church ran the government, so in a sense they were coming for liberty. However, religious freedom is one of our rights we cherish in America.
3. They came to avoid persecution.
I'm sure that reason does not appear in any family Bibles, but it was feared enough that the US government put in place more stringent immigration rules so we did not end up with all world's criminals. This did not appear to be a widespread problem at the turn of the 20th century, however, according to this paper. I'm sure there were some good stories out there, though, about folks who ran from the law.
4. They came because a relative was already here.
Often men would come first, get a job, earn money, and then send for their wives and children. Or older children in a family would come first and prepare the way. I'm sure many people have stories in their families about reunions at Ellis Island and other immigrant stations. At Ellis Island in the room where families were finally reunited, there was a post nicknamed The Kissing Post.
5. They came not to stay.
This was particularly true of some Italian immigrants in the early 20th century. They brought no family, sent for no one, and came just to work and save enough money to buy their own businesses or farms in their native country. This was the Industrial Revolution. American needed workers. These immigrants put up with squalid living conditions because they hoarded all they could to send home to their families and to invest there. It's not a story you hear very often when you look at those tenement pictures. I'm not saying everyone who came chose that kind of life, but some did. And some who had planned to only stay temporarily actually never did go home.
What stories have you heard? Why did your ancestors immigrate?