This is a far more difficult question than you might expect. Torah has no dogma, no formal set of beliefs that one must hold to be a Jew. In Torah, actions are far more important than beliefs, although there is certainly a place for belief within Torah.
The closest that anyone has ever come to creating a widely-accepted list of Jewish beliefs is Maimonides' thirteen principles of faith. Maimonides' thirteen principles of faith, which he thought were the minimum requirements of Jewish belief, are:
As you can see, these are very basic and general principles. Yet as basic as these principles are, the necessity of believing each one of these has been disputed at one time or another, and the liberal movements of Judaism dispute many of these principles.
Unlike many religions, Torah does not focus much on abstract cosmological concepts. Although Jews have certainly considered the nature of God, man, the universe, life, and the afterlife at great length, there is no mandated, official, definitive belief on these subjects, outside of very general concepts such as the thirteen listed above. There is substantial room for personal opinion on all of these matters, because as said before, Torah is more concerned about actions than beliefs.
Torah focuses on relationships:
Our scriptures tell the story of the development of these relationships, from the time of creation
The scriptures also specify the mutual obligations created by these relationships, although various movements of Judaism disagree about the nature of these obligations. Some say they are absolute, unchanging laws from God (Orthodox); some say they are laws from God that change and evolve over time (Conservative); some say that they are guidelines that you can choose whether or not to follow (Reform, Reconstructionist). For more on these distinctions, see Movements of Judaism.
Maimonides' position in Mishneh Torah is that while the core of the Law is fixed for all time, there is room for rabbinical legislation and interpretation. This is rather like seeing the God-given Written Law and Oral Law as a constitutional framework for legislation; in this case, the constitution is fixed for all time without amendments, unlike a man-made constitution, but its understanding and application are not inflexible, when a Supreme Rabbinical Court (or Sanhedrin) exists. Only this position fully fits the evidence in the ancient Oral-Law literature, in our opinion. Unfortunately, we have not had a Sanhedrin for about 1500 years, so that as a practical matter, the Law is not given to change today.
So, what are these actions that Torah is so concerned about? These actions include the 613 commandments given by God in the Written Torah as well as laws instituted by the rabbis. These actions are discussed in some depth on the page regarding Halakhah (Jewish Law) and the pages following it.