Immigration tends to be caused by a mixture of “push” factors and “pull” factors. Push factors make immigrants want to leave their home countries. Pull factors make them want to come to the US. Eras of heavy immigration to the US have typically been caused largely by push factors in foreign countries but also partly by pull factors from the US.
Push factors from other countries have tended to be political and economic. The massive wave of Irish immigration in the 1840s came about because the Irish Potato Famine made it very difficult for many people to survive if they stayed in Ireland. The influx of Jews from Eastern Europe in the late 1800s was caused largely by fear of pogroms.
The main pull factors from the US have always been economic opportunity and political freedom. These have been fairly constant over the years. During periods when there was less opportunity, however, immigration has declined. For example, the Panic of 1893 slowed immigration sharply during a time when immigration was otherwise quite high. The same thing happened after the major financial crisis of 2007-8.
Immigration, then, is typically caused by problems in the home country that push people out and the general political and economic attractiveness of the US as a new place to live.