That is a really big question, I'll try to compress an answer into something useful. You probably know some of this already but I'll start from the beginning to stay coherent.
During the late part of the Second Temple period, the three main sects were the Pharisees, Sadducees, and Essenes. To over-simplify it, the Sadducees were mostly tied to the Temple elite and their focus was on the sacrifices and such that happened physically at the Temple. The Pharisees were more populist and focused on what you are supposed to do as an individual in your day to day life, and the Essenes were monastic/mystical. After the destruction of the Temple, there wasn't much point in being a Sadducee anymore and the Essenes faded into history, though a lot of their practices were reflected in Early Christianity.
That left the Pharisees, which are the ancestors of the Rabbinic tradition. There were a few steps on the way to creating the Rabbinic tradition, but it might be useful to start out with Hillel and Shamai. They are the proto-typical frenemies that are quoted frequently in the Talmud. Hillel was the more accepting one and Shamai the more severe one. Their two groups of students argued all the time, but Hillel usually won. A Talmudic story that best shows their relationship, and is referenced all the time, goes like this (Talmud Tractate Shabbat 31a):
"There was another incident involving one gentile who came before Shammai and said to Shammai: Convert me on condition that you teach me the entire Torah while I am standing on one foot. Shammai pushed him away with the builder’s cubit in his hand. This was a common measuring stick and Shammai was a builder by trade. The same gentile came before Hillel. He converted him and said to him: That which is hateful to you do not do to another; that is the entire Torah, and the rest is its interpretation. Go study."
But, it is working backwards to call them Rabbis yet, even though they usually are. They were Pharisees. The better starting point for the Rabbinic tradition is with Yohanan ben Zakkai. During the siege of Jerusalem, he snuck out of the city and convinced Vespasian to spare "Yavne and its scholars." So, while the temple was destroyed, there was still a community of scholars to keep the religion alive. There were a bunch of scholars after that, other notable ones were Akiva and Shimon bar Yochai, who were supporters of the Bar Kohba Revolt.
The Talmud itself is made up of two parts. The first part, the Mishnah, was edited and put together by Rabbi Judah HaNasi, who lived in the immediate aftermath of the Bar Kohba Revolt. The Mishna is the Oral Torah that is traditionally believed to have been given to Moses at the same time as the written Torah. A more historicist account was that it was the accumulated oral tradition of Jewish law that had built up over the previous centuries. The Gemara is the commentary on the Mishnah by any number of Rabbis. It usually takes the form of an argument over what the Mishnah actually means, sometimes without ever actually coming to a conclusion. Most of it is about law, but there are also a lot of folk stories and such in there like the one I quoted. Pirkei Avot(Ethics of the Fathers) isn't really laws so much as advice and guidelines. There are actually two Talmuds, the Jerusalem Talmud was made in the Tiberius in the 400s and the Babylonian Talmud, the one that is used more, in what is now Iraq in 500.
If you look at a page of Talmud, you will have the Mishnah on top, Gemara on the bottom, and around the margins there will be other commentaries written by other Rabbis in the intervening centuries.
Jewish religious authority was mostly fairly centralized in Babylon in late antiquity, but by the time of the early Middle Ages it had spread out more. The most commonly read biblical and Talmudic commentator is the Rashi (Rabbi Shlomo Yitzchaki) lived in 10th century France. Others in that time period include Maimonides (Moshe ben Maimon) in Muslim-run Cordoba and Nachmanides (Moshe ben Nahman) in Catalonia.
By the time of the late middle ages and definitely by the Middle Ages, most Jews had settled into the two broad categories that exist now. Ashkenazic Judaism, which started in the Rhineland and spread eastward, and Sephardic, which started in Spain but dispersed around North Africa and the Ottoman Empire after the Alhambra Decree and Inquisition, and more or less took over the local communities where they went since Spain was already such an important center of learning. Those are the two big groups, smaller communities come from Italy, Greece, Central Asia, Persia, the Caucasus, and Yemen and do their own thing.
The next development took place within Ashkenazic Judaism. 17-18th century Poland was a rough place to be, as it often has been since then as well. Beginning with the Khmelnytskyi rebellion in 1648, the Swedish Deluge, Ottoman Wars, etc. the region was torn apart several times over and people were really open to some new ideas. So the Baal Shem Tov (Rabbi Israel ben Eliezer) began a the Hasidic movement, which took the principles of the Kaballah and made them more approachable for the average person. They were in conflict with the more traditionalist Rabbis based out of Vilnius.
Then came the Enlightenment. The Jewish version of the Enlightenment was called the Haskallah. The Haskallah was at its biggest in what is now Germany, especially Prussia. It took the humanist, rationalist ideas of the time and mixed them with Judaism. So, 19th century Germany was where the Reform Movement started, which looked at how to have a Judaism that was more modernized and compatible with being a member of a modern nation-state. Services began being conducted in German language, pipe organs, etc. The conservative movement was mostly the same. Modern Orthodoxy and similar movements like the Neologs in Hungary kept the Orthodox framework, but made more allowances for modernity.
Which is basically where we land now. You (reductively) have Ashkenazi Jews from Europe and Sephardi Jews from the Middle East and the Balkans. Orthodoxy is divided into Modern Orthodoxy and Ultra-Orthodoxy, which is itself then divided into Hasids and Litvins, which are themselves made up of dozens of smaller dynasties and traditions. Then Progressive Judaism is divided between Reform and Conservative, and to a lesser degree Reconstructionist, Renewal, and Humanist. That is all under the umbrella of Rabbinic Judaism. Karaite Judaism is a tiny, tiny minority. There used to be a pretty sizeable community of Karaites in Crimea. But then the Holocaust happened, and then after they survived that, they were also Crimean Tatars, which Stalin wasn't a huge fan of. So that community is nearly extinct.