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 to Judaism.
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 has not undergone the formal process of conversion but who believes everything that Orthodox Jews believe and observes every law and custom of Judaism is still a non-Jew, even in the eyes of the most liberal movements of Judaism, and a person born to a Jewish mother 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, Judaism is more like a nationality than like other religions, and being Jewish 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.