Israel: the religious name of the Jewish people.
The word Israel has a complex history. Originally, the expression was used to indicate one of the two kingdoms that came into being after the fall of the united monarchy of Saul, David, and Solomon, in c.931. In the south, the house of David continued to rule Jerusalem and the kingdom of Judah, and in the north, the kingdom of Israel came into being. Its capital was Samaria.
Although Israel was a powerful state, it was unable to withstand the increasing power of Assyria, and in 724, it became a province of the empire, called Sâmerîna. Part of the population was deported to the east and never returned (the well-known 'lost tribes'). Several prophets announced that Israel would be restored, but this did not happen - at least not in the way they had foreseen.
The name Israel is also used as a surname of the patriarch Jacob. This is probably a retroprojection from a later age.
The southern kingdom, Judah, was never fully occupied by the Assyrians and remained independent. Yet, after the fall of Assyria and the rise of Babylonia, Jerusalem was also captured and Judah became part of the ancient near eastern monarchy. Again, an elite was deported to the east.note[Go here for the overthrow of Assyria, and here for the fall of Babylon.]
The exiled Jews in Babylonia now started to reread the sacred scriptures. They read about the future restoration of Israel, and argued that these were not erroneous prophecies, but concerned them. They must have grown more confident with this identification when they did indeed return: in 539, the Persian king Cyrus the Great captured Babylon and allowed all exiles to go back home.
From now on, the inhabitants of Judah used the word Israel to describe themselves, even though they had always called themselves Judaeans. It is not uncommon that a nation has two names, one to describe themselves in internal discussions and one to be used in a dialogue with outsiders. For example, today, the country of the Dutch has two names: when the Dutch speak among themselves, they call their country the Netherlands, but the rest of the world knows this country as Holland.
So, in ancient sources, the Greeks and Romans call the country of the Jews Judaea and its inhabitants Judaeans. The Jews on the other hand, especially when they are discussing religious matters, call themselves Israel.
One of the most interesting examples can be found in chapter 15 of the Gospel of Mark, when the author describes the crucifixion of Jesus of Nazareth. Mark mentions that Pontius Pilate, a Roman, ordered a notice to be placed on the cross, saying that the crucified man was BASILEUS TÔN IOUDAIÔN, "king of the Judaeans" (=Jews). A few lines later, the Jews mock Jesus, and say "He saved others but can't save himself. Let this Messiah, this king of Israel, come down now from the cross, that we may believe in him."