They lived cold, hard, impoverished lives on the fringes of society. There are three main regional groups: the Ashkenazim (Europe and Russia), Sephardim (Spain and northern Africa), and Mizrahim (Middle East, northern Africa and Asia). They regarded as social undesirables, but valuable as a merchant class. Catherine the Great kept the Jews of Russia (and Poland) in the Pale Settlement and eventually these communities become educated in Western European universities and take their ideas back to their communities. While this is going on political leaders like the czars and their nobles are facing a lot of pressure because of the social inequality and economic backwardness of the land in their control. They need a scapegoat and point their finger at very culturally different Jews (many of whom collect taxes for the czar). The citizens in Russia take out their aggression on the Jews. There's a lot of murder, rape, looting, and arson going on known as the pogroms. The idea of Zionism is in the backs of many secular Jewish minds, but Theodore Herzl in France is the one individual who sparks the fire and gets things rolling after his disgust at the Dreyfus Affair. He basically lays out the idea the the Jews of the world should all live together and not have to deal with people and states that want want to kill them just because they're Jews. Long story short there is quite a bit of support for returning to Eretz Israel. The Zionist Organization led by Chaim Weizmann bought up a bunch of swampy, malaria infested land off wealthy Arab landowners who lived elsewhere (this is all taking place in the Ottoman Empire that was around until it collapsed in 1923) . I'm skipping ahead here, but the some of the people that were drawn to the Zionist movement were actually able to make it to the land of the Palestinian Mandate/Israel/Judah/milk and honey. Most of them got to live too. The ones who stayed behind in Europe were not so fortunate. The Holocaust, world guilt, refugee surplus, civil war, terrorism/freedom fighting and anti-colonialism all became arguments for the necessity of Zionism.
I don't really feel I did your question the justice it deserves but there are descriptions of every day life in novels and memoirs like: Solomon and Gaenor (this is actually a film not a novel), Everything is Illuminated, The Revolt, Fiddler on Roof, and TONS of material written in Yiddish and Modern Hebrew.
The mods will probably delete this comment, but I'm not sure what you mean, Israeli as an identity didn't really exist before Israel was founded. Do you mean Jews in general?
As a result of the diaspora the Jewish population lived all over europe. After they were kicked out of Spain by Isabella of Castille they came to mainly reside in eastern europe. They didn't find much luck there either and by the 1800s the treatment of Jews, particularly the Russian forced emigration of Jews into certain areas of the empire, encourage the growth of the Zionist ideology.
To summarise the Israeli people, if by that you mean Jews, lived throughout europe following the diaspora, but mainly in eastern europe, and were particularly poorly treated wherever they went.