Halacha
Halacha is the Jewish code of law that was derived from the Bible through the tools of exigesis and handed down, according to tradition, from Moses when he received the law at Mt. Sinai. The word, coming from the hebrew root H-L-CH, walk, is a conjugation meaning "the way," or "the path." It is supposed to be a definitive set of rules regarding the permissibility of any give action in Judaism.
Halacha is typically divided into two categories, Bein Adam L'Chaveiro, (Between a man and his fellow) and Bein Adam L'Makom (Between a man and God.) This includes everything from an understanding of what beleifs define a Jew, to who is responsible if an animal does damage, to What foods are and are not permissible to eat. Also included in the category of Halacha is the legal codes for the Temple, when it will be rebuilt, and what is ritually pure and impure, whether the laws have any practical value currently or not.
The oldest source for the Halacha is the bible, which is, on some level, the source for all of the laws. There are 13 exigetical laws to derive what the law is from the actual verses. These law include such idea as when a verse first lists a general rule, and then a specific law, the law comes to limit the application of the genral statement, whereas when the general rule follows the specific law, its means that the rule applies to any case similar to the case of the specific law, in addition to the cases where it would normally apply. Using these rules, sources are identified in the Mishna and Talmud for the laws which were handed down from Moses. Parenthetically, it is important to note that when the Mishna, then later the Talmud was written, the laws themselves, for the most part, were clear to everyone involved, and only in very specific cases is there any lack of clarity.
History
After the Bible was written, the Jews went to the land of Israel, where, except for a 70 year exile between the first and second temples, the law was kept continuously for 9 centuries. Most questions of the law during this time, of course, were trivial; monetary judgements, damages, etc. questions of whether something was ritually pure or not. These cases were dealt with by a court of three rabbis, of which there might be many in a single city, and certainly one in every town. More important cases were deferred to a court of twenty three; if a capital case were to occur, it would need to be tried in front of one of these courts, though actual death penalties were rare, and intended to be a deterrent. (Tractate Makkos, near end of the first chapter) Another reason that a case might come in front of the regional court of 23 is if a local court did not know what the rule should be. In this case, all of the judges would be polled, and if one of them remembered the rule, it would be discussed and verified, and the case would be resolved. If, however, the case remained insoluble, it would be referred to the national court of 71, located in the Temple, where the Judges would once again be polled. The other reason the court of 71 might be consulted is if one of several cases occured; If there was a rabbi who rebelled against the court (Deut. 17, 8) or if an entire city indulges in Idol worship (Deut. 21, 17). If, however, the case is truly new, and none of the judges has learned of it from their teachers, this court of 71 is empowered to interpret the law, based on their understanding of the law, and make a decision, which is duly recorded in case it re-occurs. (This is according to the Rambam's understanding of the system, where it is not clear from the Mishna, in Tractate Sanhedrin, what occurs.)
Once, however, the second temple was destroyed, and consequently the death penalty (and the punishment of lashes, which is also considered part of that category of punishments) was stopped, the courts of 23 were no longer convened. The court of 71, the Sanhedrin, lasted longer, (see Tractate Gittin for the full story of this occuurance,) and made many rulings that allowed Judaism to survive the long exile from the land of Israel. Eventually, however, this too was disbanded. For many years, the teachings were passed down from teacher to student purely orally, like it had been at the time of the Temple. Eventually, however, there was a concern that the law would be forgotten, and a set of notes were committed to writing, on which there was much unstated teachings that could be taught, still, from teacher to student. This compilation was the Mishna. In the next several centuries, as the Jews were further dispersed, there was a concern that even this knowledge was being lost, and it was decided to compile the Talmud. First the Jerusalem Talmud (sometimes erroneously referred to as the Palestinian Talmud,) then the Babylonian Talmud were compiled, from approximately 300 C.E. until 500 C.E. These recorded discussions of the time clarifying exactly what the disagreements in the Misna were, and often introduced new disagreements, or resolved how to rule in old ones.
The Babylonian Talmud was, for several centuries, the definitive understanding of what the Halacha was in any given situation, with the Jerusalem Talmud as an auxillary source, used if the Babylonian Talmud did not discuss a subject. During this time, the talmud was either the direct source of the law, or in cases that were not discussed, the direct source of the Rabbis that answered questions. These responsa are the main remaining literature from the period, and it is clear that the judgements made were based specifically on rules or logical deductions from the Talmud. After approximately five hundred years, the midevil comentators began to record their understanding of the talmud, section by section, line by line. The most renown of these commentators is Rashi, who wrote the first, and definitive commentary on how to understand the discussion in the Talmud, line by line, around the year 1100.
In the next several centuries, more commentaries were written by various Rabbis, notably (and approximately Chronologically, excuse my memory for any lapses) Tosafos, the Riff, the Rambam, the Rush, the Ran, the Ritva, the Rashba, the Tur, and Rav Yoseph Cairo. Several of these are notable in the development of Halacha. Firstly, the Riff re-wrote a version of the Babylonian Talmud containing only the parts that he felt were applicable to the Halacha, leaving out the Aggadata (see Talmud,) the discussion leading up to the conclusion that does not add to the final understanding, and any dissenting opinion. This was his life's work, and was the first major step in modern Halachic thought. (This was still in the 13th century.) Following him, the Rambam wrote the forerunner of all later Halachic works, his Mishna Torah, also known as the "Yad Chazakah." This was a new compilation of the law, organized differently than the Mishna (into fourteen sections, instead of six,) the first person to do this. It was also monumental, and unique, in that it included laws that did not apply at the time, such as the laws of sacrifices in the temple, and ritual purity. After this, the next major step towards modern Halacha was the Tur, who once again re-arranged the law, and split it into four sections. The sections are especially important because Rav Yoseph Cairo followed it when he wrote his work, the Shulchan Aruch.
The Shulchan Aruch, written around 1400, was writeen, much like the Rambam's Mishna Torah, as a definitive source of Halacha. Instead of compiling his own thoughts, however, he distilled everything up to that point. Starting with the Tur's organization, he went through and reasoned out the various understandings of the Midieval, coming to a conclusion which he stated in large part by quoting sections of the various commentator's writings, compiling it into coherent statements, so that on cursory examination you can understand to law, but with more depth, you can understand all of the intracacies of the decisions. Preference in deciding what to decide in the law is given to the Riff, the Rush, and most of all, to the Rambam. This, of course, ignores the fact that most of these explanations are in large part either discussions of what Rashi and Tosafos thought, or explanations of points they did not discuss, so that their explanations serve as a basis for all Halacha, though they were not writing specifically to explain the law.
Right before the writing of the Shulchan Aruch, a division among Jews that occured throughout the centuries became more important and pronounced; Jews in Western Europe were isolated from Jews in spain and throughout the middle east, due to the ongoing wars between Christians and Muslims. This led to an inability to have a consensus between all of the Halachists around the Jewish world. What occured was that when new situations occurred in Halacha, different decisions were made in different parts of the world.
Notably, on Passover, there is a rule that any grain that swells cannot be eaten. When rice became prevalent in the western world, the Sephardic, or Spanish Jews (and due to the strangeness of the political geography at the time, the Middle Eastern Jew as well) decided that rice did not qualify, and therefore eat rice on passover. The European Jews, however, decided that rice was a grain that counts as swelled when soaked in water, and said that it may not be eaten on passover. Similar divisions were made about corn, and several other items on Passover.
In addition, different customs were adopted regarding how to pray, such as the order of certain prayers, the Sephardim and Ashkenazim differed, enough so that typically they pray in seperate groups with their own order of prayer. Many other customs, less divisive but equally estranging, developed.
When the decree that men should only marry one wife was decided, it was only spread to the european countries, and spain, but was not really accepted in the middle east, where it did not really ever spread to. In fact, when the modern state of Israel was formed, and Jews fled from the Arab countries that ejected them, the state was confronted with many men who were married to multiple wives. It was decided that legally the marriages would remain binding, but that no men could marry additional wives from that point on.
Due to these differences, Halacha in many places is split into different groups, with the Rambam and the Shulchan Aruch being the sources for Sephardim, and Rav Moshe Eisserlis, the Rema, being the main source for Ashkenazim, writing a set of notes on the Shulchan Aruch about where the Ashkenazim disagreed with the law as stated within. Another, smaller, set of Jews followed only the Rambam's laws. There are also some differences within the Ashkenazi community, where the law that one must wait between mean and milk is interpreted four different ways; for the majority of jews, the first decision made, that of waiting six hours between them, was accepted, though there was a dispute whether this meant a full six hours, or only waiting until the sixth hour begins. The leaders of the German Jewish community decided that it meant three hours, and the leaders of the Jewish community in Holland instucted their community to only wait one hour between eating meat and milk.
Halacha Now
Since the Shulchan Aruch, it has been the focus of most Halachic commentator's work, with dozens of full commentaries written since then. However, the most important work in Ashkenazi Halacha was done in a more reductionist fashion, attempting to write more usable digests of what we rule in the most common cases. These include the Mishna Berurah, written by The Chafetz Chaim, and based on that, the Kitzur Shulchan Aruch (the short Shulchan Aruch,) discussing the daily aspects of Halacha. The Sephardi communities, having a much more definitive set of rules as laid out by the Shulchan Aruch, and having the additional distraction of much worse ongoing Antisemitism than most of European Jewry, had much less written in terms of non-responsa literature.
In addition to this, there have been many important developments, both technologically, and socially, requiring additions to, or adjustments from the standard body of law. These cases were dealt with by the Halachic leaders of the generation, in their ongoing Responsa, which form an important addition to the mainstream digests such as the Mishna Berurah. Based on the Talmud and a logical approach to the problems, issues such as whether electricity could be used on the Sabbath were dealt with. (And since it is not, was the prohibition from the fact that one cannot light fires, or from the fact that construction of a conduit for something such as water is prohibited?)
The most recent writers of Responsa, specifically those since the writing of the Mishnah Berurah, the last definitive collection, are especially important, since they deal with issues that were not compiled. The most notable of these responsa writers are, for Ashkenazim, the Chasam Sofer, Rav Aharon Cutler, and Rav Moshe Feinstein, and for Sephardim Rav Ovadia Yosef. Since that time, much of even their writing have been simplified and divided into topical treatments of various sections of the law, such as laws of Sabbath, laws of Blessings on food, etc.
In short, the law, as passed down for the last 3500 years, with a somewhat considerable effort, is traceable from the Bible, all the way to what the most recent questions of laws concerning the Sabbath. This "path" that Religios Jews follow is a transmission through the interpretation of the most erudite and revered scholars of each generation, that leads directly to the revalation of God to all of the Jewish people when he spoke directly to the nation. (Exodus 20, 1)
Note
Firstly, despite the definition of Webster below, Halacha is not a type of exposition of Midrash. This, as far as I know, is not a question of current usage, but a problem of translation.
Secondly, this writeup was written by an Ashkenazi, Orthodox Jew. If you feel that the reality is somewhat different than what was stated here, feel free to respond, but I am reasonably sure that what I have said would be agreed to by the consensus of Orthodox Jews, and while I am sorry I have not given Sephardic Halacha the attention that it deserves, in my defense, it was mostly through ignorance, not malice.