The "written" Torah, or TaNaK, contains the Five Books of Moses, Nevi'im (Prophets), and Ketuvim (Writings, or scrolls).
The "oral" Torah, or Mishna, began long before the Hebrews were taken to Babylon in captivity, ca. 586 BCE. (The word mishna comes from the Hebrew root for "repeat" because it was transmitted orally from teacher to student.) The Mishna was finally put into writing around 200 CE by Rabbi Judah HaNasi.
Discussions about the Mishna, which included the parts of the Torah from which the Mishna is derived, is called the Gemara (from the word mara, meaning to learn or study.) The Gemara is written in Aramaic, the common tongue at that time. Unlike the Mishna, which is relatively brief, the Gemara is long, convoluted, and meanders off topic - as, for example, where in the rules for blessings we find a lesson about why it is wrong to embarrass another person. Together, the Mishna and Gemara comprise the Talmud.
In the Gemara, the rabbis dissected the Mishna to determine what it said about the Torah law. Although we find 36 ways to incur the death penalty in the Torah, and many in the Mishna, the rabbis in the Gemara gave so many rules to the courts that it became almost impossible to carry out the death sentence. They said any court handing down one death sentence in seven years is a bloody court.
The Mishna contains six sections, called orders or sederim (as in the word for seder or siddur). These sections are: Zeraim (seeds) - laws about agriculture and blessings over food; Moed (times and festivals) - laws for the sabbath, holidays, and fasts; Nashim (women) - laws about marriage, divorce, contracts, and vows; Nezikin (damages) - civil and criminal law (Pirkei Avot, Ethics of the Fathers, is in this section); Kodashim (holy things) - laws of kashrut and Temple sacrifices; and Toharot (purity) - ritual purity laws.
Each Order is divided into smaller books called Tractates. The Tractates are divided into chapters. The chapters are divided into smaller numbered sections called Mishnayot. The entire Mishna is made up of thousands of Mishnayot. Some of the Mishna doesn't have a Gemara, as in Pirket Avot, but when it does, the commentary is about both the Torah portion from which it is derived, and the Mishna itself.
Because our rabbis lived in both Babylon and Judea, they discussed the Mishna in both places, hence we have two Talmuds - the Bavli and the Yerushalmi. Citations from Talmud indicate which of the two (the Bavli by default), the Tractate and folio (or daf, an Aramaic word for plank, board, or oar - because the arrangement of words on the page resembles an oar, with the Mishna at the top, the Gemara below, together forming a T-shape, and the sides filled in with comments of later medieval rabbis.)
The Talmud is the beginning of Jewish law (the Torah describes Hebraic law) because the rabbis were determining how to preserve their religion when there was no longer a Temple at which to worship. The dinner table has become a substitute for the altar. Sacrifices were replaced with prayer. Jewish law was discussed in the Gemara over a 1000-year period, from about 586 BCE to 500 CE. These laws don't appear in order, due to the meandering nature of the rabbis' conversations. Someone in one century might argue with someone from another century, and their comments will appear side by side as if they were all in the same room. The Gemara gives numerous sides to every argument - many, many different perspectives, with some decisions left up in the air until Elijah comes to provide a definitive answer.
12th century physician and philosopher Moses Maimonides, AKA the Rambam, compiled a summary of Jewish law in his book(s) the Mishneh Torah. Joseph Caro, in 15th century Spain, put together his own separate list. Each of them chose what he liked, what made sense to him, probably, I would imagine, in the spirit of offering a concise overview much as I am attempting to do here. Both books omitted the nuanced and multifaceted discussion that characterizes the Talmud, leaving us with plain legalism. When people study only the works of the Rambam and Caro, they miss out on the rest. From these men, one might learn how to kasher a plate, but perhaps not the importance of feeding the hungry.
In this series I am going to attempt to share with you what I am learning from my ongoing study of the Talmud. I would welcome the contributions of others who may also be engaged in this study process. We can find in this ancient text so many ideas that are as relevant today as when they were written.