Trigger said:
Abraham is first introduced in the bible as a "Hebrew". Ostensibly the term comes from his ancestor Eber and applies to all the descendants of Eber. But the only descendants of Eber the bible keeps track of are Abraham and his family.
Outside the bible Egyptian records refer to a people of the desert called "Apiru" which may be the biblical Hebrews but may also include other Semitic peoples of the Ancient Near East.
"Israelite" refers to the descendants of Jacob, Abraham's grandson, who was also known as "Israel". Today the only identifiable descendants of Jacob are the Jews. But this can be a religious and cultural identification, not necessarily a biological one, since Judaism does receive converts.
When the descendants of Jacob were divided into two groups, "Israel" was the name given to the northern kingdom in which the largest and most influential tribe was Ephraim. Occasionally, in this sense, the kingdom of Israel is called Ephraim or even Joseph (father of Ephraim). The southern kingdom was called Judah, since this was the principal southern tribe. The peoples of Simeon and Benjamin were absorbed into the kingdom of Judah. From this point on "Israel" and "Judah" refer to political entities rather than blood relationships.
Still later, the Israelites (northern kingdom) were conquered by the Assyrians, and much of the population was deported and replaced by other peoples. And a further century later Judah was conquered by the Babylonians and much of the population deported. Israel as a political entity ceased to exist, but Judah managed to survive exile and return to its traditional territory under the Persians. It was never again independent (except for a brief period under the Maccabees), but was a province of various empires (Persian, Greek, Seleucid, Roman) until the destruction of Jerusalem in 70 CE.
"Jew" is derived from "Judah" but no longer refers specifically to descendants of Judah, but rather to those who identify with the community returned from exile whatever their tribal origin.
Personal histories are separate from political histories. Even at the time of the split between the north and south, many northeners who remained loyal to the covenant migrated south, so the kingdom of Judah included people of many tribes, not just the tribe of Judah. Others migrated into Judah when the kingdom of Israel was conquered, and from time to time people of Judah migrated north. And since the Babylonian and Persian empires covered much the same territory as the Assyrian empire, citizens of Israel and Judah probably intermingled in exile as well. There were also Jewish exiles in Egypt and various other parts of the Mediterranean. All of these, whatever their family and tribal roots, eventually identified themselves religiously and culturally with the Jews of Judah/Judaea.
It is this blend of ethnicity, religious belief and culture to which "Jew" refers today.