What is the principle of Tribal Sovereignty? Does it mean that Native American tribes are not part of the United States?
Native American tribes are part of the United States, but have a special status in a lot of circumstances. Tribal/reservation land is under federal jurisdiction, but they have their own government for day to day operations. Tribes are recognized by the U.S. as independent, but they lack a real level of sovereignty from the federal government.
Native Americans have been considered U.S. citizens since 1924. The Indian Civil Rights Act (1968) granted many of the constitutional rights to those within Tribal jurisdiction that had previously been ruled to not apply.
Voting rights are still to some degree a contested topic, Native Americans have had the right to vote since the Voting Rights Act, but this has angered States with larger Native populations because those natives aren't under state jurisdiction.
Hunting and Fishing rights is the most controversial and significant difference, and tends to be the source of most anti-Native American sentiment. Native Americans are permitted to hunt/fish on Tribal lands or lands that were historically tribal lands with no permits or restrictions, this includes privately owned lands and use of modern commercial fishing practices.
Taxation is another big issue, but it is a much more complex issue that would require another answer to go into full detail. There is some level of taxation that those living on tribal land are exempted from.