The Hawaiian chiefdoms were simple chiefdoms meaning that there was not that many complex hierarchical institutions. The chiefdoms were built around the idea of kinship and the chief would usually be the oldest member of a select family or a number of families which were something like aristocracy and they would rule the chiefdom.
The first source of power was mana. Mana was a spiritual concept where everyone has Mana in themselves, and Mana can be found in anything else as well. All your actions either increase your mana or reduce your mana and there were several points that were considered to be strongly imbued with Mana.
The second was ideology. Chiefdoms usually supported everything that the chief said. Religion and political relations were important in this, especially considering that various types of deviant behavior were considered to be bad because the gods would disapprove them. This was a great mechanism of control and keeping the tradition alive.
The third was military. It was fairly simple, the Chief was in control of the military. If anyone decided to go against the chief they would have to suffer because the military would stand and fight. Although the military served to protect the people from other chiefdoms or other enemies, it was another mechanism of control of the population.