Today, the Cherokee Nation's government is based very much on the government of the United States or of the various states. That is, it is a democracy with separation of powers and three branches of government. As an Indian nation, the Cherokee have some degree of sovereignty within the United States.
The Cherokee legislature is called the Cherokee Nation Tribal Council. It is made up of 17 members. 15 are elected from within the nation's "tribal service jurisdiction" (the nation does not have a reservation) and two are elected by Cherokee living outside that area.
The Cherokee also have an executive branch with a Principal Chief and a Deputy Principal Chief at its head. These are elected positions as well. Finally, there is a judicial branch that is not elected.