What are the sources for the Jewish law or "Halacha"? Is there any priority among these sources? In common practice, when a law is described, is the citation given form the main sources or derived codified books? (Please consider explaining any Jewish/Hebrew terms.)
G-d gave all the laws to Moses, and he passed it on to his students orally until the time of R' Yehudah, who wrote everything down in the mishnah in a concise way (Gitin 60a). We accept everything written there, since it came from G-d Himself. However, by that time, disputes had already risen over things that had to do with logic (which we are expected to apply to other cases when relevant), but not over the tradition itself (Tosafos Yom Tov, Avos 1:4). At around the same time, baraysos (laws not included in the mishnah) were written by contemporaries of R' Yehudah (e.g. Tosefta by R' Chiya). (See also this.)
R' Yehudah had three students, Rav, Shmuel, and R' Yochanan (Introduction to Mishneh Torah). Rav and Shmuel moved to Babylonia, and are therefore major authorities in the Talmud Bavli, while R' Yochanan moved to Israel, and became an authority for the Talmud Yerushalmi (these are both commentaries on the mishnah). However, they didn't actually write these Talmuds, but rather their students did. It turned out that the Yerushalmi was completed before the Bavli. Therefore, we assume that the ones who put everything in the Bavli saw everything in the Yerushalmi, so if the Bavli disagrees, we assume the Bavli disagrees even though it saw the Yerushalmi, so it must have had a good reason for disagreeing. However, if the Yerushalmi is not in doubt and the Bavli is in doubt, we follow the Yerushalmi. This is the explanation that I heard. See also Sdei Chemed Klalei HaPoskim 2:1.
Also, there was the Zohar, written by R' Shimon bar Yochai, but we follow the Bavli over Zohar (Mishnah Brurah 25:42 in the name of the Kneses HaGedolah; see also these sources). We also follow Yerushalmi over Zohar (see this answer). We follow Tosefta over Yerushalmi as well (see Sdei Chemed Klalei HaPoskim 2:5; the Pri Chadash quoted there disagrees). There were other works (e.g. Sifra, Sifri, etc.), but I won't cover them all.
Afterwards, disputes arose in the intent of those Talmuds themselves. There rose up many works of law in order to clarify which opinions in the Talmuds we follow, etc. There were many works (e.g. Bahag), but the Rif's commentary was the first most accepted original work to explain the law. After him rose the Rambam, who most of the time followed the Rif (Migdal Oz in many places). The Rosh also wrote a commentary. These three works were the ones that formed the basis for most of the laws of the Tur (the son of the Rosh).
The two people who wrote the first commentary on the Tur were the Beis Yosef (R' Yosef Karo) and the Darkei Moshe (R' Moshe Isserless), who explained (and sometimes disputed) the decisions of the Tur. But after all their comments, it ended up being that it took a lot of work to find the practical halachah. Therefore, the Beis Yosef wrote the Shulchan Aruch ("set table"), arranged like the order of the Tur, which only contained the practical halachah, its source being the corresponding chapter in the Tur, from which anyone could see how the law developed.
After the Shulchan Aruch, new commentaries came up to explain it and dispute it, so now the most common work to find the halachah is the Shulchan Aruch and its commentaries.
(As a side note, these commentaries which disputed their predecessors didn't dispute the authority of the Talmuds; they either found a case in the Talmuds which they brought as proof against a different case, or made rulings based on logic.)
Traditional Judaism regards the oral law as the primary means of interpreting the written law - i.e. the Pentateuch and the rest of the Old Testament. The oral law is a combination of specific laws which the Tradition says were transmitted by God to Moses at Sinai and a code of various methods of exegesis by which to derive laws from the Bible.
This second aspect is where disagreement mainly begins; since these new laws are being filtered through human reasoning, sometimes one person says one thing should be derived and another says something else. This level of disagreement is commonplace in the Mishna and Talmud, which are basically compendiums of the Oral law and the arguments which developed it. [You might also find a disagreement here and there about whether or not a particular tradition is accurate, though this isn't as common.]
Halacha as it is today is basically the final word of the Talmud on the subject. We accept the Talmud's word as to what is or isn't a valid derivation or law and we generally do not attempt new derivations from Scripture; rather we fully submit to the authority of the Talmud. Disagreements today are of a different sort - they are disagreements about how to apply Talmudic law to the endless amount of cases not specifically talked about in the Talmud. Also, there are sometimes disagreements about what the conclusion of Talmud was; in places where it may be ambiguous.
To answer your questions:
A final note: Many times when talking about Halacha someone will cite the Shulchan Aruch or other earlier opinions. It isn't that the Shulchan Aruch is on the level of the Talmud; it is really just R. Yosef Karo's opinion of how to rule on practically every matter that existed in his day. However, it has been accepted by the Jewish community for the last few hundred years as the basic code of Jewish law, and so while theoretically it is acceptable to disagree with its rulings, doing so is usually seen as a very strong departure from precedent. The same is true with regard to Rema's glosses on the Shulchan Aruch, for Jews of Ashkenazic descent. For this reason you will often hear people simply citing these and similar works without tracing all the way back to the Talmud, as these works generally give a good indication of normative Halacha which itself ultimately traces back through the entire process.