This is a far more difficult question than you might expect. Judaism has no formal set of beliefs that one must believe. In Judaism, actions are far more important than beliefs, although there is certainly a place for beliefs.
The closest that anyone has ever come to creating a widely-accepted list of Jewish beliefs is Maimonides's 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 other religions, Judaism does not focus much on abstract concepts. Although Jews have certainly considered the nature of G-d, man, the universe, life and the afterlife at great length (see Kabbalah and Jewish Mysticism), there is no mandated, official, definitive belief on these subjects, outside of the very general concepts discussed above. There is substantial room for personal opinion on all of these matters, because as I said before, Judaism is more concerned about actions than beliefs.
Judaism focuses on relationships: the relationship between G-d and mankind, between G-d and the Jewish nation, between the Jewish nation and the land of Israel, and between human beings. Our scriptures tell the story of the development of these relationships, from the time of creation, through the creation of the relationship between G-d and Abraham, to the creation of the relationship between G-d and the Jewish people, and forward. 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 G-d (Orthodox); some say they are laws from G-d 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.
So, what are these actions that Judaism is so concerned about? According to Orthodox Judaism, these actions include 613 commandments given by G-d in the Torah as well as laws instituted by the rabbis and long-standing customs. These actions are discussed in depth on the page regarding Halakhah (Jewish Law) and the pages following it.
