The word "Jew" comes from a transliteration of the Greek Ioudaios, which is the word for Judea. It is defined as someone of the Judean nation, or of the culture of Judea, such as being associated with the temple.
At the time of Jesus, the word was used for people both of the house of Judah and the Judaized converts. Major early conversions happened during Judah's Babylonian captivity as recorded in Esther 8:17.
The Jews in Babylon had married and had children with non-Israelites, but turned them away as was recorded in Ezra 10. This left a significant base of 'Jews' which included all the converts and those that did not return to Judea.
Judea began converting when Hycanus conquered Idumea and converted the Edomites which Jospehus recorded in his works Antiquities, 13:324. He also made note in Wars 2:8:2 of the three major sects of Jews: Sadducees, Pharisees, and Essenes, only the isolated Essenes were fully Jews by birth. King Herod himself was an Edomite.