What are the essential requirements of conversion to Judaism, such that if one of these requirements is not fulfilled the conversion is invalid?
Someone wishing to convert will work with a rabbi, who will guide the convert through the process and ensure that the following requirements are met before proceeding to the beit din (court).
A convert, first and foremost, must accept the yoke of the mitzvot (kabbalat ol mitzvot). In other words, a convert must be doing it out of a sincere desire to join the Jewish people and take on its responsibilities. Rabbis will initially discourage converts.
Some course of learning will be required. From what I understand, this is a combination of nuts and bolts (b'rachot, t'filah, shabbat, kashrut, some Hebrew, etc), Tanakh, history, and theology. Once someone converts he becomes liable for transgressing mitzvot, so he needs to know how to live as a Jew first. In the US today this learning tends to be a combination of classroom study and individual study with a rabbi.
Some lifestyle changes may be necessary, ranging from replacing kitchen utensils and kashering to changing work schedules to, possibly, moving into an observant community within walking distance of the shul. The rabbi working with the convert will provide guidance here.
The ritual requirements are (per R. Maurice Lamm, Becoming a Jew, p119):
Going before the beit din, a court made up of three observant adult male Jews (in liberal movements, this may include women). Usually the beit din is made up of rabbis.1 The beit din will question the would-be convert on motives, knowledge, secular/family impact, and other topics.
If accepted by the beit din: for men, circumcision or, if already circumsized, hatafat dam b'rit, drawing a symbolic drop of blood from there.
If accepted by the beit din: for everyone, immersion in a mikvah (ritual bath). (My understanding is that for male candidates the beit din witnesses, and for female candidates they stand outside the door and rely on the mikvah attendant to confirm complete immersion.)
When the temple stands, the convert is also liable for a korban (offering).
A conversion might not be accepted by other communities, depending on who the supervising rabbi and members of the beit din are. See this question for more information.
My understanding is that once a conversion is done by a qualified beit din, it stands even if something happens to call the convert's motives into question. If it turns out that the beit din was not qualified, I'm not sure what happens -- I think there is disagreement.
1 R. Lamm (p133) says that one of the three must be a rabbi, and that rabbi chooses two other knowledgeable and observant men.
- Sincere intent to accept the Written and Oral Torah. (Some would add: to the best of their understanding.)
- For men: circumcision, or drawing blood if already circumcised.
- Immersion in a mikvah before a panel of three Jews who are eligible to render halachic testimony.
(And add an animal sacrifice, back when those were doable.)
- have a reason you feel the need to convert or to at least explore the idea of doing so. find a sponsoring rabbi who will be your teacher and guide through this whole process.
- acceptance of all Jewish beliefs and a rejection of any beliefs from other religions. a good source is the 13 principles of faith by the Rambam
- acceptance of all of the commandments and understand how they relate to daily life as well as the Sabbath and holidays
- see a bes din who will decide whether to approve your request
- bris mila or hatfas ha-dam habris for males
- Bring a sacrifice to the Temple (besides hamikdash) when it is rebuilt, may it happen very soon.