There are hundreds of recognized tribes in the US. And while there are similarities across many of them, they can be very different too. Broadly speaking tribes fall into one of two categories: traditional or constitutional. Traditional tribes are structured according to their "old ways" and usually "run" by a cacique (religious leader) for all internal tribal business and run by a Tribal Council for interacting with non tribal entities. In my experience with the 13 tribes in NM (and 1 in Washington state) Internal politics determine who gets to be on the Tribal Council. Tribes in NM (and elsewhere) are usually divided into clans (2-4) that often have competing interests within the internal politics of the tribe. Most of how all that works is kept secret from outsiders as is clan membership. Leaking internal traditional practices to non Indians usual means that if the outsider is an employee, they will be fired. Very often in traditional tribes there are predefined roles for members and lines that don't get crossed. In traditional tribes in NM women never have leadership roles and never are on Tribal Councils. Their role is to cook and support their men. They can be influencers in the background. This link goes to Sandia Pueblo a very traditional tribe of fewer than 500 members. You will notice there is next to nothing about the tribe's internal governance. Sandia Pueblo is the richest tribe in NM with well over $1 billion in investment assets. Their wealth comes from their casino on the northern edge of Albuquerque. I worked there for 6 years.
www.sandiapueblo.nsn.us
Constitutional tribes have open elections for most of the important positions and often women are allowed to participate and even hold office. Laguna Pueblo is a constitutional tribe in NM. Here is a link to their website showing election details. There is less secrecy in the constitutional tribes but none the less, much of the internal politics of those tribes are publicly known.
www.lagunapueblo-nsn.gov