What is the difference between the terms “Hebrews”, “Israelites”, and “Jews”?

All of these terms are dependent upon ancestry, rather than upon where anyone lives. An “Israelite” is any biological descendant of Jacob. Jacob was renamed “Israel” by God in Genesis 32:28. “And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God and with men, and hast prevailed.” Jacob was the father of the twelve tribes of Israel whom he blessed in Genesis 49:1-28. Thus, the nation of Israel is often called “the children of Israel” in the Bible text, because they are the descendants of the man, “Israel”.
The term “Hebrews” is almost identical to “Israelites”, but it slightly broader, because it includes the fathers, Abraham, Isaac, and Jacob, in addition to all of the descendants of Jacob. We know this because Abraham was referred to as “Abram the Hebrew” in Genesis 14:13. Therefore all Israelites are also Hebrews.
In explaining the term “Jews”, it is beneficial to understand some of Israel’s national history. After the reign of King Solomon over all the tribes of Israel around 1000 BC, the kingdom was split into two kingdoms as recorded in 1 Kings 12:1-24. The southern two tribes of Judah and Benjamin comprised the kingdom of Judah. The northern ten tribes continued to be called the kingdom of Israel. Because of the split, there are some occasions where the term “Israel” is used only in reference to the ten northern tribes instead of all twelve. But usually, the term “Israel” refers to all twelve tribes, depending upon the context.

About 300 years after the split of the kingdom, in 712 BC, the Assyrians attacked the northern kingdom and took the ten tribes of the kingdom of Israel captive (2 Kings 17:6). The earliest occurrence of the word “Jews” in the Bible is in 2 Kings 16:6, near the time that Israel was carried away. The term “Jews” refers to those of the kingdom of Judah, and their descendants. About one hundred years later, in 606 BC, the Babylonians took the kingdom of Judah captive and carried them away to Babylon (2 Kings 24-25). Seventy years later, the Medes and the Persians conquered Babylon (Daniel 5) and released the Jews. However, most of the Jews chose to remain in Babylon, and only a small remnant (Ezra 2:64) of about 42,000 returned to the land of Israel.
Paul said in Acts 21:39, “… I am a man which am a Jew of Tarsus (which was not in the land of Israel) ….” He also said in 2 Corinthians 11:22, “Are they Hebrews? so am I. Are they Israelites? so am I ….” So all Jews are also Israelites and Hebrews regardless of whether or not they live in the land of Israel.
As for the northern ten tribes of Israel, the Bible does not record them returning from the Assyrian captivity. Even though their whereabouts may be unknown to men, God still knows where they are and will use them during tribulation, just like the two tribes of the southern kingdom (Revelation 7:1-8). But during the first century when the New Testament was written, the vast majority of the Israelites referred to in scriptures were from the two southern tribes. Thus, they are called “the Jews”. But even though “Jews” refers primarily to those of Judah and Benjamin, the term is probably not intended to exclude other Israelites who were living among them such as Hanna from the tribe of Asher (Luke 2:36) or Barnabas from the tribe of Levi (Acts 4:36).
In Acts 26:2-7, Paul defended himself before King Agrippa saying, “… I shall answer for myself this day before thee touching all the things whereof I am accused of the Jews: Especially because I know thee to be expert in all customs and questions which are among the Jews: wherefore I beseech thee to hear me patiently. My manner of life from my youth, which was at the first among mine own nation at Jerusalem, know all the Jews; Which knew me from the beginning, if they would testify, that after the most straitest sect of our religion I lived a Pharisee. And now I stand and am judged for the hope of the promise made of God unto our fathers: Unto which promise our twelve tribes, instantly serving God day and night, hope to come. For which hope’s sake, king Agrippa, I am accused of the Jews.” Notice here that Paul seems to be implying that “our twelve tribes” are of the Jews, which would seem to support the idea that the term “Jews” had taken on a broader meaning by the first century, where it no longer meant exclusively the tribes of Judah and Benjamin.