how does the bible define Jewish?

The word Jew, until about 100 years ago was an English derogatory term to insult Jews.

Thus "Jewish" is not defined in any Jewish sources.

"Jew" and "Jewish" is used by Christian Bibles where the translators were being hateful to Jews.

The polite term by non-Jews was "Hebrew"

The polite term is still is "Hebrew" in most languages other than English.

This is used throughout the Bible --

starting from Abraham (it means "the ones from across the river" -- and Abraham's family were named that by the Canaanites)

and used in the story of Joseph

and in the story of Jonah

The term used by Jews internally is "Israel"

That's why the Jewish country was named Israel as soon as it obtained full Independence.

"Children of Israel", of course, starts into use with the 12 families of the sons of Jacob (Jacob also known as Israel), and becomes the national name when the Jewish nation state is created in the time of Moses.

The originally insulting English term "Jew" was derived from the term "Judea" applied by Rome and Greece ("Iudea" actually) to refer to the Kingdom of Israel.

(from Rome to the Catholic Church - and hence all Christian cultures)

This term first came into use to refer to Israel by the Babylonians.

It originated as a nickname of one of the two half-kingdoms of Israel - one was Yehuda and the other Ephraim. But by the time the Babylonians invaded, Ephraim had been destroyed by Assyria.

So, in the Babylonian period, all Israelis came to be also nicknamed Iudeans.

The term "Judaism" was created by Greek historians ("Iudaismus" in Greek) who wrote about Israel.

Iudaismus refers to the system of Israeli laws and religion and culture.

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> Is it Hebrew heritage, culture, or religion?

Yes.

The old polite terms in English were

"The Hebrew people"

"The Hebrew language"

"The Hebrew religion"

"The Hebrew culture"

...only the term "Hebrew language" remains part of general normal English usage.

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> People say Jewish is a culture-but what does that mean exactly-celebrating religious holidays, or there are secular aspects to it? Are the secular aspects derived from the religion?

The distinction between "secular" and "religious" did not exist when the nation of Israel was established.

The Torah ("The Laws of Moses") was the Constitution of the nation of Israel and is believed by the Hebrew religion to be a covenant with God.

It includes regulations of religious behavior (sacrifices, holidays)

as well as "secular" laws - negligence, assault, fraud, murder, ...even public health regulations like creating a latrine far enough away from the camp to prevent dysentery and cholera.

As is the case for almost all words used in the Bible

the Bible does not define the word "Jewish".

They are a people with one religion

They have been know as the people of God, Israelites, Hebrews, and finally Jews ( a name derived from the Kingdom of Judah which was the good half of the the crowd when the kingdom split into the ten tribes of the northern kingdom called Israel and the South Kingdom of Judah (Comprised of Judah and Benjamin)

Technically the word "Jewish" applied only to those who lived in the Kingdom of Judah which was the southern kingdom after the split. "Israelite" refers to those from the northern kingdom which was called the Kingdom of Israel.

Circumcision (traditionally), but you can be a Jew at heart as well.