The Iroquois led a half-nomadic life until they came to the NE woodlands. There they discovered a novel way of agriculture.
By interplanting corn, beans (both originating from the souuth) and local squash, they significantly increased agricultural output while keeping the soil fertile for a very long time. This allowed them to settle down in permanent (20-30 years and more) towns.
Other tribes lived in the same area (using more primitive methods like slash & burn), but at the time there was plenty of space.
Still they quarreled a lot (too much free time, I think

) with their neighbours and among themselves, until Hiawatha and the Peacemaker made an end to most of that by forming the League in 1451 AD. From that moment the Iroquois prospered even more.
When the Europeans arrived, they were in the best position to control the fur trade routes. The French-supported Algonquin and Huron attacked them with guns, but when they got their own from the Dutch they were able to build a large empire, cleverly replacing their fallen with anyone who was willing to swear by the Great Law of Peace. Other tribes, such as the Sioux, were driven away onto the plains or even put out of existence (remnants being adopted).
They suffered from the European diseases, but not as much as other tribes, since they never knew hunger. But eventually they overextended, getting mixed up in too many European battles.
When, because of conflicting promises, they found themselves on two sides of the American war of Independence, this spelled the end of their empire and they lost most of the territory they had conquered. But while their empire ended, many of their ways and ideas remained, such as their democracy and respect for women, and eventually had a huge impact on the entire world.
Meanwhile, the Iroquois were never conquered themselves, and the Confederacy is still in existence. They still have their own law and government, flag and passport. Their numbers today are larger then ever, it is only because of the huge mass of immigrants and their descendants that they seem to have grown insignificant.
But even today they are not. Iroquois are advising environmental movements, making essential contributions by trying to teach a world of spoilers how to survive their own waste.
The Mounted Warrior is, obviously, a stupid choice by Firaxis for a civ that lives in the woodlands. The proper UU for the Iroquois would IMHO be a Musketman that treats all terrain as roads.
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The Cherokee are Iroquoian, one of the closest cousins of the Iroquois.