How to become a Jew is perhaps almost as contentious as the question of “Who is a Jew?” because the two questions are not mutually exclusive. The one informs the other.
Halakhically – that is, according to Jewish law – one becomes a Jew when one follows a certain process. This involves:
1. Learning about Judaism with a rabbi. This is not as easy as it may sound. Learning the history of the Jewish people is indeed important. The history of Jewish people will hopefully teach the potential convert what the future may be for his/her potential future people and himself/herself – he/she learns what he’s/she’s getting into. But this is only part of the picture. One must also learn the many rules, rituals, and laws that will pertain to the potential Jew. Examples of issues one must become familiar with are the rules of kashrus, the Jewish calendar, the Jewish festivals (and the rituals in each), Jewish prayer, the rules on ritual purity (taharah), and the rules of “family purity” (taharas hamishpokha).
2. The potential convert must indicate he/she knows what he/she is getting into, and evince a continuous desire to become a part of the Jewish people.
3. The potential convert, once prepared, must be examined by a panel of rabbis (beys din), who will ask questions to test him/her, and to make sure he/she is truly accepting the yoke of Heaven (qabbalas ‘ol malkhus shamayim). If he/she passes, the panel approves he/his conversion.
4. Men must undergo circumcision (bris milah if they are not circumcised, or undergo a symbolic bloodletting (<hatafas dam milah).
5. The convert must immerse in a mikvah (tevilah b’miqvah) to become ritually pure.
After all the requirements are met, the beys din will issue a certificate certifying the person’s conversion. The convert is also given (or chooses) a Hebrew name. (For various purposes, Jews use a formula of (Hebrew name) son/daughter (ben/bas) of (father), and sometimes (Hebrew name) son/daughter of (mother). The latter is used mostly for prayers of healing. In such a case, converts will use “Avraham” (Abraham) for the father’s name and “Sarah” for the mother’s name, indicating their spiritual descent from the Patriarch and Matriarch.)
There is also a tradition among Orthodox rabbis to thrice reject a person’s request to begin the process of converting to Judaism. This is to ensure the person really, really wants to become a Jew.
Now, the contentious issue is under whose authority can or ought one to convert? The government of Israel does not recognize conversions done under non-Orthodox rabbis. (In fact, the situation has become worse: the government of Israel recognizes only those US conversions that were under the auspices of a certain list of Orthodox rabbis.) Conversion under Orthodox Judaism is harder, longer, and more intensive. While Conservative and Reform Judaism relaxes some of the requirements, Orthodox Judaism does not. And so the issue remains: are people who convert under a Reform rabbi not really Jews? The Orthodox would say that would be the case. The Reform Jews, understandably and obviously, would be highly offended and would disagree. Orthodox Judaism doesn’t have a monopoly on the definition of a Jew. However, this is what it seems the government of Israel is saying.
In any case, this is how the conversion process works in Judaism.