The first one was 585 to 515 BCE.

the second is - 70 CE to the present day.

• Answer 2

The Diaspora (scattering of the Jewish people) began because we were unable to live in the Holy Land. The prophets had constantly warned the Israelites that ignoring the Torah would result in exile.
In 68 CE, the Romans destroyed the Second Temple. The Romans did not force the Jews out of Judea in a single expulsion. Rather, the Romans expelled them from Jerusalem only; and the rest of Judea lost its Jews slowly, over a period of centuries, as living there became too harsh.
Those Jews who left Judea went to southern Europe, North Africa, Arabia, the Near East, and (slowly) further afield (especially throughout Europe).