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The Mountain Clans are tribes of people who live outside of the whole social system of the Seven Kingdoms and usually stay in the mountains near the Vale of Arryn. They are the medieval equivalent of living off of the grid; except, instead of doing business just in cash and not having any bank accounts, the Mountain Clans do a bunch of raiding. So it's no surprise that they're considered dangerous barbarians by everyone else.
What is a surprise is that Tyrion is able to convince several of these clans to join him in fighting for the Lannisters. That just goes to show how awesome Tyrion is: he meets a bunch of people who threaten to kill him, and by the end of the negotiation, Tyrion is leading them to war. Well done.
We don't really get to see a lot of the Mountain Clans, but we do learn a few things about them. For instance, they all practice democracy, which Tyrion finds bizarre (57 Tyrion 7.9). There are some cultural differences between the various tribes: the Stone Crows, the Black Ears, the Burned Men, and the Moon Brothers (we call dibs on all of those for band names).
Though we don't spend too much time with them, a few of the Mountains Clan people stand out from the crowd (the crowd, in this case, is wearing a lot of fur from animals they killed). For instance, the Stone Crows' chief, Gunthor, agrees to stay back in the mountains to recruit for Tyrion. Shagga, another Stone Crow, is a giant who enjoys threatening to feed people's body parts to goats. Chella of the Black Ears is very proud of how many ears her sons have taken in battle (57 Tyrion 7.70). This one's interesting: Timettof the Burned Men put out one of his eyes as his adulthood ritual. Everyone agreed this was pretty impressive, so they made him an important person (a Red Hand), even though his depth perception is terrible. Finally, the Moon Brothers send along a tribeman named Ulf to make sure that Tyrion keeps his word.
Hopefully that guide will help you keep the Mountain Clans straight. But the important thing to remember is that the Mountain Clans prove how smart Tyrion is and help protect him during the battle of 63 Tyrion 8.