Depending on the context, one or some or all might count for an individual. It looks like the JFS is sticklers when it comes to needing an orthodox rabbi to do the conversion.
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It is NOT a race nor ethnicity because there are Jews of every ethnic group.

You can change your religion and no longer be considered Jewish.

You cannot change your ethnic background.

Here’s an experiment: look up photos of Yaphet Kotto, Alyson Hannigan, Paul Newman, Adam Sandler, Natalie Portman, Lauren Bacall and Harrison Ford

Are Jews one ethnic group? Obviously not – all of them were born Jewish.
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There used to be a race of the Jews, but it is now called Israelite
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See:

Who is a Jew?
A Jew is any person whose mother was a Jew or any person who has gone through the formal administer of conversion to Judaism.

It is vital 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 administer 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 schedule 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. See What Is Judaism?

This has been established since the earliest days of Judaism. In the Torah, you will see many references to "the strangers who stay among you" or "righteous proselytes" or "righteous strangers." These are innumerable classifications of non-Jews who lived among Jews, adopting some or all of the beliefs and practices of Judaism lacking going through the formal administer 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.

Although all Jewish schedule agree on these general principles, there are occasional disputes as to whether a particular individual is a Jew. Most of these disputes fall into one of two categories.

First, traditional Judaism maintains that a person is a Jew if his mother is a Jew, regardless of who his father is. The liberal schedule, on the other hand, consider a person to be Jewish if either of his parents was Jewish and the child was raised Jewish. Thus, if the child of a Jewish father and a Christian mother is raised Jewish, the child is a Jew according to the Reform movement, but not according to the Orthodox movement. On the other hand, if the child of a Christian father and a Jewish mother is not raised Jewish, the child is a Jew according to the Orthodox movement, but not according to the Reform movement! The matter becomes even more complicated, because the status of that children’s children also comes into question.

Second, the more traditional schedule do not always acknowledge the validity of conversions by the more liberal schedule. The more modern schedule do not always follow the procedures required by the more traditional schedule, thereby invalidating the conversion. In addition, Orthodoxy does not accept the authority of Conservative, Reform and Reconstructionist rabbis to perform conversions, and the Conservative movement has debated whether to accept the authority of Reform rabbis.

More here:

http://www.jewfaq.org/whoisjew.htm
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1. Practicing the Jewish religion
2. having Jewish ancestry
3. being a member of a culturally Jewish family
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