The Red Power movement was called that as a takeoff on the "Black Power" movement. While the Black Power movement was about empowering African Americans, the Red Power movement was about empowering Native Americans.
Native Americans were unhappy with the general policies of the federal government up to that time. Since the 1940s, the federal government had been pursuing a policy called "termination." This was meant to try to break up the Indian tribes and assimilate the people as individuals into American society.
In addition to this policy, Native Americans were unhappy with general conditions on reservations. Native Americans were poorer, unhealthier, and less educated than any other group. These were the sorts of conditions that the Red Power movement was meant to combat.