So I know the Mongols became a powerful force under Genghis/Chingis, I know that a splinter group founded the Yuan Dynasty of China, and I know there were a bunch of other khanates/hordes/groups of Mongols who existed at the same time and were mostly led by Genghis' descendants (well, I'm not 100% sure of that last detail). But what were the extents of these groups? Which one(s) were supposedly on the brink of rampaging through Europe? After that 'invasion' failed were they mostly a spent force, or did they do something else interesting? I believe one of them was instrumental in setting up the unification/creation of the Russian state at one point?
I realise this is a pretty broad topic, but I'm mostly looking for an overview. If I can get a handle on the basics hopefully I can do more research myself.
Golden Horde were the ones on the western frontier in what is now parts of Russia. They were given to Jochi (Genghis' oldest son) after his death. however, Jochi died before his father so they were given to his sons, one of whom was Batu Khan. The reason that there were so many different 'splinter' groups is because they were split among all of Genghis Khans sons. However, ogedei, Genghis' third son was the chosen heir. So he was like the overlord of all the different khanates and he ruled the homeland.
The golden Horde never really launched any massive failed invasions so I'm not sure what you are talking about. Really their falling apart was largely due to internal struggles.
After the death of Genghis the empire stayed together as a unified state for some time. But it was divided into sections, each governed by one of his sons. These territories became functionally hereditary and Genghis' family divided into royal houses based around this first generation (Ögedei, Jochi, Chagatai, and Tolui).
So nearest Europe you had the the territory that had been given to the house of Jochi. Jochi died before Genghis, and it was being run by Genghis' grandson Batu, who was the function head of the house of Jochi. They eventually, when the empire broke up became The Golden Horde.
After Genghis the next Great Khan was Ögedei. It was during his reign that the big invasion of Europe took place. Russia was conquered. And Hungary was mostly conquered, and Poland and Silesia was pretty beat up, when Ögedei suddenly died. The Mongols withdrew from Hungary, but not from Russia.
And they went back to select a new Great Khan. There had been general agreement about Ögedei. He had been selected by Genghis. And the other brothers had agreed. This time around there wasn't a clear agreement. And there was some bad blood. Batu was able to delay the selection for several years, but eventually Ögedei's son Güyük was selected as Great Khan.
This was a problem for Batu. As he and Güyük hated each other. And they were both gearing up for a civil war. And then Güyük suddenly died.
Batu was able to then swing the next selection to one of Tolui's sons, Möngke. There was some outrage that it had been taken from the house of Ögedei. And there was even a serious coup attempt. But Möngke was able to hold it together. And the empire reached its greatest extent under him. They conquered most of the rest of the Middle East, and most of the rest of China. And they even raided into Delhi and Vietnam.
And then Möngke died. And the Mongols withdrew again to select a leader. It never quite worked out. The Mamelukes in Egypt beat the small force that had been left nearby, it was not the whole force that had been advancing on them. And the Mongol empire broke up into civil war. Möngke's brother Kublai was eventually sort of recognized as the next Great Khan, but that was an empty title, as no one took directions from him. And a generation later, Kublai's grandson Temür Öljeytü was able to force the other Mongol states to recognize him as Great Khan. But Möngke was the last Khan who could actually command the whole of the Mongols to do something and actually get it done.
After Möngke's death. The empire broke up. Ariq Böke and Kublai, Möngke's younger brothers fought for the throne. With all the royal cousins playing a role. And powers as far removed as the Franks, the Mamelukes, and the Japanese got involved in one way or another. The house of Ögedei, out of power since the coup attempt was particularly active... and faired particularly poorly.
The Golden Horde was being ruled by Berke, Batu's brother. It is rumored he had Batu's sons killed. He had converted to Islam, and so was not happy with the conquest of Muslim lands. He favored Ariq Böke. But only really hated Hulagu.
The Ilkhanate was being ruled by Hulagu, Möngke's brother. One of Hulagu's wives was Christian, and he was generally easy on Christians and harsh on Muslims. He is the one who sacked Baghdad and destroyed perhaps the greatest Muslim city in history. He and Berke were at each other's necks almost immediately on Möngke's death. He was Kublai's only constant ally.
The Chagatai Khanate filled in that central part of Asia. Alghu (Chagatai's grandson) wasn't the ruler when the civil war broke out. But he used it as an excuse to gain power and functional independence. He supported everyone at one point or another finally siding with Kublai, but with the understanding that he would be more or less independent for having done so.
The Yuan Dynasty was founded by Kublai after the civil war, and is counted from this point as a "proper" Chinese dynasty. Kublai kept China, Mongolia, Korea... but never got more than nominal rule over any of the rest again.
And that is about as short and simple as I can make it.