The main power that the Supreme Court of the United States has is the power of judicial review. This is the power to rule that actions of the federal government or of state governments are unconstitutional. This is a very significant power in that the Court has used it many times to strike down laws made by one part of government or another. This has led to such important things as the end of racial segregation and the legal right to an abortion.
However, this is not the Court’s only power. In general, the Court has the power to hear appeals from a variety of courts. (It also has the right to hear a few kinds of cases that come through its original jurisdiction.) It hears appeals from lower federal courts as well as from courts of last resort at the state level.
The Supreme Court has the right to decide these appeals based on statutory law or on constitutional law. When the Supreme Court rules, there is no court in the land that can overrule it.
It is important to note that the Supreme Court lacks the power to legislate or to enforce its decision. The Court cannot strike down government because they are bad ideas. It can only strike them down if they are contrary to the Constitution or to statutory law. Because of this, the judicial branch has been seen as the “least dangerous branch” of the federal government.