By the first century CE, the term rabbi (“my master”) became common to refer to a learned Jewish teacher. Also by that time, competing factions existed in Judaism: the Pharisees, Sadducees, and Zealots. An ascetic group, the Essenes, also lived during this time, said to be successors of the Zadokite priests that began in the times of David and Solomon. Here is an explanation of differences among these groups: http://www.jewishvirtuallibrary.org/pharisees-sadducees-and-essenes Early Christianity emerged during this time as well.
After the First Jewish–Roman War (66–73 CE) and the Roman destruction of the Second Temple in 70 CE, Jews found themselves in a traumatic situation that rivaled the disaster of 586 BCE. How would the faith survive without a temple? Without a priesthood? How would Jewish traditions survive? The Zealots all died at Masada in 73; the Sadducees faded from history; and the the Essenes also disappeared. The Christian sect became a predominantly Gentile religion, retaining Jewish scripture and reconfiguring aspects of Jewish theology. Jewish vitality remained with the Pharisees, who saw Jewish law as the focus of Jewish life, and they helped shift the focus from temple offerings to tzedakah, study, and synagogues.
In this situation, a tradition called the Oral Torah had to be compiled and written down. The Written Torah was the scriptural five books of Moses, but an oral tradition attributed to Moses’ teachings had been passed down over the centuries; Orthodox Jews believe that this tradition was safeguarded through the Judges and Prophets and Second Temple-era sages. After the unsuccessful revolt of Simon Bar Kochba (132-136 CE), Romans forbade Jews from returning to Jerusalem---further exiling Jews, and further necessitating a way to preserve Jewish faith and traditions. By about 200 CE, Pharisaic Judaism had segued into Rabbinic Judaism as a rabbi named Judah ha Nasi (Rabbi Judah the Patriarch) began to edit these oral traditions and discussions about Jewish law into a readable form during the early 200s CE.
The first compilation and written/edited form of the Oral Torah is called the Mishnah. Rather than a law code, it is a study book (or rather, several books) containing the varieties of discussions and opinions of the sages. Rabbi Judah drew from many sources in his compilation and recorded discussions in a way to help with memorization. These are by no means uniform opinions. If the sages differed on when morning prayers should begin, what defines a Jewish marriage, and many other topics, the differences are recorded. Here are some sample passages: https://people.ucalgary.ca/~elsegal/TalmudMidrash/MishnahSamples.html
The Mishnah has six orders (sedarim): agriculture (Seder Zeraim), sacred times (Seder Moed), women and personal status (Seder Nashim), damages (Seder Nezikin), holy things (Seder Nodashim), and purity laws (Seder Tohorot). Each order is divided into tractates, and each tractate has chapters, and each chapter contains halakhot (laws) of the Tannaim, who were the sages from the era of the Mishnah (like Rabbis Akiva, Hillel, Shammai, and many others: see this site).
The word Tosefta means “addition,” and the Tosefta is a body of material that further explains Torah laws, details about laws, and provides extra material to the Mishnah. The Tosefta is three times as large as the Mishnah, although it is also structured with six orders. There are different theories as to whether the Tosefta is older than the Mishnah and was originally and independent body of opinion, or whether it was compiled and written later in order to broaden the material of the Mishnah, which does not include rabbinic discussions preserved in the Tosefta. Editions of the Babylonian Talmud provide the Tosefta at the end of each tractate.
What is the Talmud? Talmud is the comprehensive collection of the Oral Law that encompasses the Mishnah (200s CE) and the Gemara (500s CE). Talmud is discussion of the Mishnah but also the Mishnah itself.
The word Gemara (from the word gamar study) refers to the rabbinic commentary discussions about the Mishnah. You could say that the Talmud is the Mishnah plus Gemara, with material from the Tosefta as well. The sages of the Gemara (the period 200-500 CE) are referred to with the term Amoraim (see this site); the Amoraim expounded on and explained the Oral Law transmitted by the earlier Tannaim.
The site “My Jewish Learning” has this: “Although it is organized in accordance with the structure of the six orders of the Mishnah, mishnaic teaches are, for the Gemara, the launch pad for diverse topics: prayer, holy days, agriculture, sexual habits, contemporary medical knowledge, superstitutions, crumble and civil law. The Germara contains both Halakhah (legal material) and Aggadah (narrative material). [My emphasis] Aggadah includes historical material, biblical commentaries, philosophy, theology, and wisdom liberature. Stories reveal information about life in ancient ties, among Jews and between Jews and their neighbors, and folk customs. All of these genres are blended together with the halakhic material, in what is sometimes described as a stream-of-conscious fashion filled with meaningful tangents and digressions… [T]he Gemara … explains unclear words or phrasing [in the Mishnah]… provides precedents or examples to assist in application of the law and offers alternative opinions from sages of the Mishnah and their contemporaries [Tannaim]. Whereas the Mishnah barely cites biblical verses, the Gemara for every law discussed introduces these connections between the biblical text and the practices and legal opinions of its time. It also extends and restricts applications of various laws, and even adds laws on issues left out of the Mishnah entirely…. Multiple opinions of sages are weight against one another, often without presenting a conclusion.” myjewishlearning.com/article/gemara-the-essence-of-the-talmud
There are two versions of the Talmud; the second and later one is the more comprehensive. Scholars of the Land of Israel (especially the Galilean cities of Tiberias and Caesarea) published what is now called the Talmud Yerushalmi (Jerusalem Talmud) during the period 350-400 CE. Unfortunately, Constantine, the first Christian Roman emperor, persecuted Jews and the Jerusalem Talmud remained incomplete. Meanwhile, scholars at Jewish academies in Sura, Pumbedita, and Mata Mehasia published their own discussions in about 500 CE: this material is called the Talmud Bavli, or the Babylonian Talmud. Usually, the words Gemara and Talmud refers to the Babylonian Talmud. The language of the both Talmuds are dialects of Hebrew and Aramaic.
Long after the Amoraim, rabbinic commentators continued to discuss the law and the opinions of the sages, and so the Talmud was never a “finished” body of work. Not surprisingly, it is a vast work, running several volumes, and has been translated into English. Here is a site that provides the Bavli in Hebrew and English translation: https://www.sefaria.org/texts/Talmud Notice how it's organized according to the Mishnah sedarim that I listed above: agriculture (Zeraim), sacred times (Moed), women and personal status (Nashim), damages (Nezikin), holy things (Kodashim), and purity laws (Tohorot).
Different bodies within Judaism today view the Talmud differently. To generalize: Orthodox Jews consider the Oral Torah as inspired and authoritative, of Mosaic origin; Conservative Jews also honor the sanctity of Oral Torah and view Talmud as complementary to Torah study; Reform Jews retain Talmud studies in rabbinical seminaries but do not consider the Talmud as binding today.
Another, smaller body of material is the Pirkei Avot (“Ethics of Our Fathers”), a text which is often published separately and found in many prayer books, and which has inspired its own commentaries. Technically, the Pirkei Avot is part of the Mishnah, specifically the ninth tractate (with six chapters) in the Seder Nezikin, which in turn is the fourth order of the Mishnah. The Pirkei Avot is popular because it provides ethical principles of the rabbis and give us a sense of who they were and their devotion to Torah. “The worldview espoused by the rabbis quoted here emphasizes learning, service of God, discipleship, ethical behavior, humility, and fair judgment… A rabbi is introduced, often, but not always, as a disciple or son of the preceding rabbi, and the text then offers one or more teachings by this rabbi” (myjewishlearning.com/article/pirkei-avot-ethics-of-our-fathers/ )
For all of this material, I relied upon the helpful articles at the site My Jewish Learning (myjewishlearning.com). Subsequently I made a donation to the site. More detailed articles on the Mishnah and Talmud can be found at jewishencyclopedia.com and jewishvirtuallibrary.org/mishnah. My grateful thanks to an esteemed Jewish friend and colleague who read and commented on the essay; any remaining errors are mine.
Here is a wonderful statement that explains the devotion to study (which, after all, is based on words of God, Deut. 6:7): http://www.aish.com/atr/Why_Study_Torah.html