The year was 1905, and as seen in the musical, it was a particularly bad year to be Jewish in Russia. Also, WWI would begin in Europe in 1914 and the Tsar would be killed/overthrown in 1918.
"Pogrom" is a word we hear many times throughout this show and it means an organized killing of a minority. Many dictionaries will actually attribute Tsarist Russia in their definition of this.
We cannot know for certain, considering a great deal of this play deals with uncertainty. Optimists may think of Tevye and his family in America being able to avoid the horrors of WWI, while pessimists may focus on the impoverished Motel and Tzeitel who may not have been able to leave Europe in time to avoid WWI. Many people have taken a stab at what they think may have happened. Alexandra Silber (who played Tzeitel in the 2015 revival) wrote a book called After Anatevka, which is a sequel in which Hodel becomes the main character. I personally find this a very interesting perspective to look into.
The Tsar was the supreme leader of Russia and in 1905, it was Nicholas II. He was the last of the Romanovs to be rulers in Russia. In 1905 (the same year as this play) he had thousands of peaceful protesters gunned down in what later became known as "Bloody Sunday". They were protesting low pay, and poverty is one of the many themes in Fiddler. He also called for many pogroms against Jewish people between 1901 and 1906, calling for them to be killed/ removed from their homes. He also did a poor job of leading Russian troop in WWI, and this lead to his people murdering him in 1918.
References:
http://fordcenter.blogspot.com/2010/10/history-in-fiddler-on-roof.html
https://www.biography.com/royalty/nicholas-ii
http://alexandrasilber.squarespace.com/after-anatevka