The Supreme Court hears cases involving the Constitution of the United States. Normally it has appellate jurisdiction, meaning that it does not decide the facts of a case, but has the power to accept or overturn the rulings of lower courts on the basis of law. It has the authority to invalidate laws created by state or federal government if those laws are found to violate the Constitution. In practice this is the most important role of the Supreme Court; in validating or invalidating state and federal laws it has had a large impact on national policy ranging from campaign finance regulations to LGBT rights. Famous cases such as Roe v. Wade and Brown v. Board of Education are almost always cases where the Supreme Court exercised appellate jurisdiction to rule a law Constitutional or un-Constitutional.
The Supreme Court does occasionally have original jurisdiction, meaning that it decides the actual facts of a case as a lower court normally would. This is true when dealing with suits between two or more states, as well as with any cases involving international diplomats or treaties.
While the Supreme Court will only hear cases that are referred to it by lower courts, the Supreme Court can refuse to hear any case (with a few rare exceptions), and typically accepts only 100 to 150 of the 7,000 cases it receives each year.