Which Sovereign Has Jurisdiction Depends on Many Factors
Indian law lives in the peaks and valleys of three different sovereigns: the United States, the State of Washington, and the tribe. Per the doctrine of discovery, conquest, and the U.S. Constitution, the United States is the supreme sovereign.
Congress, therefore, can assert its power over the Reservation whenever it wishes. For example, Congress passed the Major Crimes Act in 1885, which asserted federal jurisdiction over murder and other crimes. The act has subsequently been amended and supplemented with other laws several times.
If you are an Indian and you commit a major crime, you will be subject to federal law, rather than tribal law, and your trial will be in federal court. If there is no federal law on point, the federal law will “borrow” the law of the state where you allegedly committed the crime.
Many other federal laws, for example, the requirement to pay income tax and laws regarding workplace safety and environmental protection also extend to the Indian tribes. Cases and controversies involving these laws can be heard in federal court.
State laws, for the most part, do not apply to Indian tribes and therefore state court will usually not be available to hear disputes that commonly arise between Indian tribes and non-Indian tribes, for example, employment disputes, or torts committed by a tribal employee against a non-tribal member at a casino.
That leaves the tribes. The default rule of thumb is that a tribe has jurisdiction over tribal members, but not over non-members. The exceptions are where the non-member has entered into a consensual relationship with the Tribe or where lack of jurisdiction over the non-member would compromise the political integrity, economic security, or health and welfare of the tribe.