American Indian civil rights are somewhat different than civil rights for other Americans because of the dual nature of the rights held by Native Americans. For most Americans, civil rights consist of those rights that are enjoyed by virtue of American citizenship. These rights are spelled out in the Constitution. American Indians are (since 1924) all citizens of the United States and so they have these civil rights just as everyone else does.
What sets Native Americans apart is the fact that they are legally also members of sovereign tribes. Because these tribes are sovereign, their members also have rights that come not out of the Constitution but out of treaties made between the United States and those tribes. Therefore, American Indians also civil rights such as fishing rights that derive from these treaties.
It is important to note that tribal governments must also respect American Indians' civil rights under the Constitution even though those governments are sovereign. This was mandated by the Indian Civil Rights Act of 1968.