Judaism is not a race
Judaism is not a race because Jews do not share one common ancestry. For instance, Ashkenazi Jews and Sephardic Jews are both "Jewish." However, whereas Ashkenazi Jews often hail from Europe, Sephardic Jews often hail from the Middle East. People of many different races have become Jewish over the centuries.
Judaism is not a nationality
Although today Israel is often called the Jewish homeland, being Jewish is not a nationality because Jews have been dispersed throughout the world for almost two thousand years. Hence, Jews come from countries all over the world.
Judaism is both a cultural and religious identity
Being Jewish means that you are part of the Jewish people, whether because you were born into a Jewish home and culturally identify as Jewish or because you practice the Jewish religion (or both).
Cultural Judaism includes things such as Jewish foods, customs and rituals. For instance, many people are born into Jewish homes and are raised eating blintzes and lighting shabbat candles, but never step foot inside a synagogue. A Jewish identity is automatically bestowed on babies of Jewish mothers (according to Orthodox and Conservative Judaism) and of Jewish mothers or fathers (according to Reform Judaism). This Jewish identity stays with them throughout life even if they don't actively practice Judaism.
Religious Judaism includes the beliefs of the Jewish religion. The way a person practices the Jewish religion can take many forms and partially for this reason there are different movements of Judaism. The main denominations are Reform, Conservative, Orthodox and Reconstructionist Judaism. Many people who are born into Jewish homes affiliate with one of these branches, but there are also those who do not.
If a person is not born Jewish, s/he can convert to Judaism by studying with a rabbi and undergoing the process of conversion. Merely believing in the precepts of Judaism is not enough to make someone a Jew. They must complete the conversion process in order to be considered Jewish. Though the different branches of Judaism have varying requirements for conversion, it is safe to say that the conversion process is very meaningful for whomever decides to undertake it.