Answer:
   
   Moses Maimonides (Rabbi Moshe ben Maimon, usually referred to in
   Hebrew by the acronym "Rambam") was one of the towering figures in
   medieval intellectual and religious life. In addition to his law code,
   he excelled in the fields of philosophy, science, medicine, exegesis
   and communal leadership. Though born in Spain, in his youth his family
   fled religious persecution, settling in Egypt. Maimonides' literary
   output includes: a work on philosophical logic; an Arabic commentary
   to the Mishnah; an enumeration of the 613 precepts of the Torah; the
   Mishneh Torah law code; the Arabic philosophical treatise The Guide of
   the Perplexed; and many letters and responsa addressed to various
   Jewish communities.
   
   One of the Rambam's legacies is what has been come to be called the
   "13 principles of faith". These are not related to any particular
   observance; rather, they are intended to map out the borders between
   Judaism and other belief systems (such as Christianity and Islam). Why
   is this necessary? There are certain laws that apply to our
   relationships with "apikursim" (from the Greek "epicurean"), minim
   (heretics), kofrim (deniers) and mumarim (non observant). The first
   three are defined by belief, so Maimonides wanted to outline the
   borders between acceptable belief systems, and people in these three
   classes. According to Maimonides (see Laws of Repentence 3:6-9), these
   people, while members of the Jewish nation, aren't believers in
   Judaism. This has halachic import, such as whether they can be counted
   toward a quorum (minyan) for prayer; whether one can share their wine,
   etc. It also has metaphysical import: believers in Judaism (including
   non-Jews who observe the Noachide covenant) are guaranteed a world to
   come; these people are not. A min (a term also used in the Talmud to
   refer to early Christians) is one who diverges on the basics of
   theology: polytheists, deists, atheists, those who believe one should
   worship G-d via demigods (middle-men), and those who say that god has
   a body. [According to the Rambam's Guide, the latter is a form of
   polytheism. He sees it as just a verbal difference between talking
   about one god who has parts and one pantheon of multiple gods.] The
   word apikoreis is the Aramaic for Epicurean, as in "eat, drink and be
   merry for tomorrow we may die" and "nothing exists but atoms and the
   void". Looking at Maimonides' code, he defines "apikoreis" as one who
   holds any of the following:
    1. There is no prophecy
    2. Moses' didn't have a special kind of prophecy (since it was Moses
       who actually conveyed the rules of behavior, both ours and
       Noachide); or
    3. G-d doesn't know what people do.
       
   Note that these are related to whether G-d's existance imposes
   requirements on human behavior (which is why the word relates to
   Epicurus). Kofrim are those who deny the divine origin of even a
   single verse of the Torah, or deny the origin of the Oral Torah, and
   those who say that some part of the Torah was later superceded. So, in
   summary: the wrong view of G-d makes one a min, the wrong view of how
   G-d relates to human behavior makes one an apikoreis, and disbelieving
   part of the Torah makes one a kofeir. Maimonides took these rules and
   to compose his 13 articles. So, the point of the articles is to give a
   rational basis to believing that Jewish observance was actually given
   to us by G-d.
   
   The RAMBAM's 13 principles, as expressed in the Artscroll Siddur
   (pages 178-180) are as follows:
    1. G-d's Existence
    2. G-d is a complete and total unity
    3. G-d is not physical
    4. G-d is eternal and the First Source
    5. Prayers should be directed to G-d
    6. G-d communicates with man
    7. Moses' prophecy is unique
    8. The entire Torah is G-d-given
    9. The Torah is unchangeable
   10. G-d knows man's thoughts and deeds
   11. Reward and punishment
   12. The Messiah will come
   13. The dead will live again
       
   Some other places to find a more detailed statement of the principles
   are as follows:
     * The original, from Maimonides' commentary on the Mishnah:
       [5]http://members.aol.com/LazerA/13yesodos.html
     * The Ani Ma'amin liturgical version (shorter):
       [6]http://www.panix.com/~jjbaker/AniMaamin.html
     * The Yigdal liturgical poem (even shorter, and might compromise
       precision in the langauge for poetry):
       [7]http://www.panix.com/~jjbaker/Yigdal.html
       
   It would take volumes to explain what these mean, but a good
   "catechism" of Jewish beliefs is the Handbook of Jewish Thought by R'
   Aryeh Kaplan.
   
   See Also: [8]Section 3.36. Torah: What is the Mishneh Torah (Yad
   Ha-Hazaqah , Sefer Mehoqeq)?