Israelites. "Hebrews" (Ivrim) means descendants of Eber (Ever). Ever was an ancestor of Abraham (Genesis ch.10-11) and the earliest Hebrews were Abraham's uncles and cousins for several generations back. They were among the Western Semites and lived in northern Mesopotamia, near the confluence of the Balikh and the Euphrates.
Abraham (18th century BCE) was called a Hebrew (Genesis ch.14) because of his wider family.
Poetically, however, Abraham himself is called Hebrew because that name (Ivri) also translates to "the other side." Abraham was figuratively on "the other side" since he was the only monotheist (Midrash Rabbah 42:8) until his teachings took root. His ancestors and cousins had slipped into idolatry well before his time, as is evident from Genesis 31:30, 31:53, and Joshua 24:2. For that reason, Jews do not bestow on them the honorific title of ancestors despite the genealogical connection.
We credit Abraham as our first ancestor despite knowing exactly who came before, since it was Abraham who founded our beliefs. Thus, "Hebrews" is often used to mean Abraham and his Israelite descendants, instead of his wider family. In this sense it can refer to the Jewish people.
The word "Hebrews" can continue to refer to Abraham's descendants until the lifetime of Jacob. After that, we prefer "Israelites," since Jacob was given that name by God (Genesis ch.35), and it is considered a national title; one of honor. "Israelites" refers to the people (Jacob's descendants) down to the Assyrian conquest (133 years before the destruction of the First Temple), some 2600 years ago.
"Jews" refers to the people from the end of First Temple times, up to this day, because after the Assyrian conquest the Israelites who remained in the land were (and are) mostly from the Israelite tribe of Judah, and the land was then called Judea. But all the above terms are occasionally interchanged.
In modern usage, we prefer to use the term "Hebrew" only to refer to the language.
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The Israelites were sometimes called "Jews", "Hebrews"(Genesis 14:13,Jeremiah 34:8+9, Deuteronomy 15:12), "the sons of Israel","the house of Israel", "the people of Israel","men of Israel", "the state of Israel" (Genesis 32:32; Matthew 10:6; Acts 4:10; 5:35; Ephesians 2:12)
There are number of other names for Jews, but most of them are impolite or outright derisive. The acceptable ones include: Children of Israel, Modern Israelites, and the Chosen People.