What is Emunah?

Emunah is too big a subject for this kind of medium. Still, here are a few sources that might help get the discussion going:

Sefer Hachinuch (Mitzva 25)
"(There is a mitzva) to believe that the world has one God Who brought everything into existence, and through His power and desire everything that ever was, is or will be came to exist. And He took us out of Egypt and gave us the Torah... (as it is written) 'Who took you out' (Ex. 20:2), that is to say: do not fool yourselves into thinking that your exodus from Egyptian slavery along with the plagues was coincidental, rather, you must know that I took you out of My own free I promised your fathers...

"The root of this mitzva requires no explanation. It is obvious that emunah is the very foundation of the religion...

"Emunah involves establishing for oneself that this is true and that anything else is impossible. And if someone is asked (about his views), he should always answer that this is the faith of his heart and that he cannot accept any other way, even if they threaten to kill him. For this readiness to act on ones attitude strengthens and establishes a firm faith of the heart.

"The rules of this mitzva include our need to believe that He is all-powerful and that all strength and His, and that we lack the strength and understanding to grasp and relate His greatness and goodness, and to avoid associating with Him any weakness or failing... All these matters are explained in the books (written by) experts in theology."

Rabbi S.R. Hirsch

Rabbi Hirsch refused to use "faith" or "belief" as a translation for emunah. "Knowing" God just because someone else told you about Him (what else, after all, does "I believe this is so" mean?) could hardly justify such an important lifestyle choice. We know God, according to Rabbi Hirsch, through the trust that develops from our experience with God's power and reliability. And emunah expresses itself through willingly allowing our lives and fortunes to develop in accordance with His Will (see Hirsch's commentary to Gen. 15:6).

Here's one of the many passages where Rabbi Hirsch discusses this idea:

The Nineteen Letters (Rabbi S.R. Hirsch), Letter Fifteen (page 200 Feldheim, Elias edition)

"You speak of dogmas. Dogmas of faith? Judaism knows 613 duties, but no commandments of faith. Those truths that form the basis of Judaism it reveals as facts, and it proclaims them for all who are able to perceive the truth. It thereby opens a broad field for thorough investigation, so that we may acquire a more profound understanding of the essence and interrelationship of creatures, the world, man and humanity, and of the evolution of history and the Divine plan regarding it. Our conclusions about the nature of all things must be derived from observation, from experience and from the Torah; the Torah must always be included with the given facts, for surely true speculation does not mean to construe worlds out of some inner notions, while keeping eyes and ears closed to the real world. True speculation takes nature, man and history as facts, investigating them in order to arrive at knowledge. To these, Judaism adds the Torah, for it is equally a fact, just like heaven and earth."

What must we believe? It's not all there is to say on the subject (and there are alternative positions), but here are the Rambam's famous thirteen principles:

Maimonides' Thirteen Principles Of Faith

Summarized from Rambam's introduction to the final perek of Mesechte Sanhedrin - not to be confused with the popular poem printed in many siddurim.

The Basic Jewish Beliefs page from discusses many of the same ideas that you'll find here, but from a different perspective.