At the heart of halakhah is the unchangeable 613 mitzvot that G-d 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 L-rd your G-d"), 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 L-rd, your G-d 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 the 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. Agricultural laws only apply within the state 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.
Sources: Judaism 101