the short of it: The Laws given from bible god to the Israelites at Mt. Sinai via Moshe (Moses--hence Mosaic) instituting the covenant between that G-d and the Jewish people.
The Long of it:
The 613 Mitzvot
At the heart of halakhah is the unchangeable 613 mitzvot that God gave to the Jewish people in the Torah (the first five books of the Bible). The word "mitzvah" means commandment. In its strictest sense, it refers only to commandments instituted in the Torah; however, the word is commonly used in a more generic sense to include all of the laws, practices and customs of halakhah, and is often used in an even more loose way to refer to any good deed.
Some of the mitzvot are clear, explicit commands in the Bible (thou shalt not murder; to write words of Torah on the doorposts of your house), others are more implicit (the mitzvah to recite grace after meals, which is inferred from "and you will eat and be satisfied and bless the LORD your God"), and some can only be ascertained by Talmudic logic (that a man shall not commit incest with his daughter, which is derived from the commandment not to commit incest with his daughter's daughter).
Some of the mitzvot overlap; for example, it is a positive commandment to rest on the Sabbath and a negative commandment not to do work on the Sabbath.
Although there is not 100% agreement on the precise list of the 613 (there are some slight discrepancies in the way some lists divide related or overlapping mitzvot), there is complete agreement that there are 613 mitzvot. This number is significant: it is the numeric value of the word Torah (Tav = 400, Vav = 6, Resh = 200, Heh = 5), plus 2 for the two mitzvot whose existence precedes the Torah: "I am the LORD, your God" and "You shall have no other gods before Me". There is also complete agreement that these 613 mitzvot can be broken down into 248 positive mitzvot (one for each bone and organ of the male body) and 365 negative mitzvot (one for each day of the solar year).
The most accepted list of the 613 mitzvot is Maimonides' list in his Mishneh Torah. In the introduction to the first book of Mishneh Torah, Maimonides lists all of the positive mitzvot and all of the negative mitzvot, then proceeds to divide them up into subject matter categories. See List of the 613 Mitzvot.
Many of these 613 mitzvot cannot be observed at this time for various reasons. For example, a large portion of the laws relate to sacrifices and offerings, which can only be made in the Temple, and the Temple does not exist today. Some of the laws relate to the theocratic state of Israel, its king, its supreme court, and its system of justice, and cannot be observed because the theocratic state of Israel does not exist today. In addition, some laws do not apply to all people or places. Most agricultural laws only apply within the Land of Israel, and certain laws only apply to kohanim or Levites. The modern scholar Rabbi Israel Meir of Radin, commonly known as the Chafetz Chayim, has identified 77 positive mitzvot and 194 negative mitzvot which can be observed outside of Israel today.
Gezeirah: A Fence around the Torah
A gezeirah is a law instituted by the rabbis to prevent people from accidentally violating a Torah mitzvah. For example, the Torah commands us not to work on the Sabbath, but a gezeirah commands us not to move a object only used to perform prohibited work (such as a pencil, money, a hammer), because someone handling the implement might forget that it was the Sabbath and perform prohibited work.
It is important to note that from the point of view of the practicing Jew, there is no difference between a gezeirah and a Torah mitzvah. Both are equally binding. The difference is just in the severity of punishment: a Torah violation of the Sabbath is punishable by death, while a rabbinical violation of a gezeirah is punishable by whipping.
Another difference between a gezeirah and a mitzvah is that the rabbis can, in rare appropriate circumstances, modify, or abrogate a gezeirah. Rabbis cannot change the Torah law that was commanded by God.
Takkanah: A Law Instituted by the Rabbis
Halakhah also includes some laws that are not derived from mitzvot in the Torah. A takkanah is a law that was instituted by the rabbis. For example, the "mitzvah" to light candles on Chanukkah, a post-biblical holiday, is a takkanah. The practice of public Torah readings every Monday and Thursday is a takkanah instituted by Ezra.
Some takkanot vary from community to community or from region to region. For example, around the year 1000 C.E., a rabbi instituted a prohibition of polygyny, a practice clearly permitted by the Torah and the Talmud. It was accepted by Ashkenazic Jews, who lived in Christian countries where polygyny was not permitted, but was not accepted by Sephardic Jews, who lived in Islamic countries where men were permitted up to four wives.
A takkanah, like a gezeirah, is just as binding as a Torah mitzvah.
Blah blah blah, if ya ask me....but works for some people.