Who is a Jew?
A Jew is any person whose mother was a Jew or any person who has gone through the formal process of conversion in full compliance with Jewish law.
It is important to note that being a Jew has nothing to do with what you believe or what you do. A person born to non-Jewish parents who believes everything that Orthodox Jews believe and observes every law and custom of the Jews is still a non-Jew, even in the eyes of the most liberal movements of Judaism, while a person born to a Jewish mother (regardless of who the father is) who is an atheist and never practices the Jewish religion is still a Jew, even in the eyes of the ultra-Orthodox. In this sense, being a Jew is more like a nationality than like other religions; it is like a citizenship.
This has been established since the earliest days of Judaism. In the Torah, you will see many references to "the strangers who dwell among you" or "righteous proselytes" or "righteous strangers." These are various classifications of non-Jews who lived among Jews, adopting some or all of the beliefs and practices of Judaism without going through the formal process of conversion and becoming Jews. Once a person has converted to Judaism, he is not referred to by any special term; he is as much a Jew as anyone born Jewish.
About Matrilineal Descent
Many people have asked me why traditional Judaism uses matrilineal descent to determine Jewish status, when in all other things (tribal affiliation, priestly status, royalty, etc.) we use patrilineal descent.
The Torah does not specifically state anywhere that matrilineal descent should be used; however, there are several passages in the Torah where it is understood that the child of a Jewish woman and a non-Jewish man is a Jew, and several other passages where it is understood that the child of a non-Jewish woman and a Jewish man is not a Jew.
In Deuteronomy 7:1-5, in expressing the prohibition against intermarriage, G-d says "he [i.e., the non-Jewish male spouse] will cause your child to turn away from Me and they will worship the gods of others." No such concern is expressed about the child of a non-Jewish female spouse. From this, we infer that the child of a non-Jewish male spouse is Jewish (and can therefore be turned away from Judaism), but the child of a non-Jewish female spouse is not Jewish (and therefore turning away is not an issue).
Leviticus 24:10 speaks of the son of an Israelite woman and an Egyptian man as being "among the community of Israel" (i.e., a Jew).
On the other hand, in Ezra 10:2-3, the Jews returning to Israel vowed to put aside their non-Jewish wives and the children born to those wives. They could not have put aside those children if those children were Jews.
