Tribal sovereignty means the ability of a Native American Nation to govern its own affairs. In practice, federally recognized tribes often function like another State (think Alabama, Vermont, Michigan, etc.) and sometimes function like a separate country (think Mexico, Canada, France). This means that tribes, just like states and countries, can establish taxes, set citizenship and voting laws, punish people for crimes, and make other laws, rules, and regulations.
Just like like France can only enforce its laws within France, Tribes can only enforce their laws on Tribal land.
Additionally, if France tried to enforce its criminal laws against American citizens in the U.S., this would be a violation of America's sovereignty. Similarly, if a State tried to enforce its criminal laws against a Tribal citizen within a Reservation, it would be a violation of the Tribal Nation's sovereignty.
Finally, it's important to note that Tribal membership is not a racial matter. Status as a Tribal Member is a political designation, not a racial one; it signifies that someone is a member of a certain nation. Simply being ethnically Native American is not enough for Tribal Membership. Each Nation has its own requirements to legally gain membership, just like the United States has its own process to legally become a citizen.
Native American Nations existed long before the United States. During the early days of the United States, the U.S. Government treated Native Nations like foreign countries. They made treaties, waged war, and recognized tribes as separate governments. Over time, as the United States expanded westward, the government forced Native people to move and make room for settlers. These forced removals were brutal, and resulted in the deaths of many, many Native people. This was what many refer to as the "trail of tears." The U.S. enacted these removals by entering into treaties with Native Nations, in which the Nations agreed to move west in exchange for a land of their own. For many tribes in Oklahoma, these treaties set up boundaries, within which each Nation could have a permanent homeland. Today, we call those permanent homelands "Reservations." Within those Reservations, Tribal Nations could exercise their governmental powers until Congress decided to terminate the Reservation. Tribal sovereignty exists because Tribal Nations have always existed as governments, and the U.S. Congress has continued to respect this.
Tribes use their sovereignty to do many of the things a typical government does. Within their reservations, Tribes fund roads, bridges, schools, healthcare, law enforcement, courts, parks, and invest in local businesses. Tribes also fund other member services to help their citizens afford homes, get career advice, repair credit, and improve their physical and mental well-being. Additionally, Tribes invest in cultural education and preservation projects, ensuring that our ancestors are honored and that our cultures and language endure for future generations. In short, Tribes use sovereignty to invest in infrastructure, health, business, education, culture, and public safety, just like any good government.
The Bureau of Indian Affairs from the U.S. Department of the Interior has prepared a good list of answers to frequently asked questions! Click the button below to see that page.