The Diaspora (the exile) started in 586 BC because the first Temple was destroyed by the Babylonians. The Babylonians then captured the Jews and brough them into exile. Between 66 and 135 AD, there were Jewish revolts against the Romans. The Second Temple was destroyed and another wave of the Diaspora came, with Jews scattering all through the world.
Eventually some of these Jews returned to Palestine, and Zionism sprang up at the end of the 19th century. The Zionists wanted the creation of a Jewish nation. In 1947, the United Nations voted to create a Jewish state, and on May 14, 1948, Israel declared independence as a state.