The Supreme Court has exclusive powers and official jurisdiction over "all cases affecting ambassadors, other public ministers and consults" as well as any case that involves a state as one of the parties in the case and a foreign government. According to the Federal Judicial Center, the Supreme Court usually exercises its power of exclusive jurisdiction when two states are in a suit against one another.
One example of the Supreme Court exercising its exclusive jurisdiction can be seen in the Rhode Island v. Massachusetts case in 1838 that dealt with "a claim of sovereign immunity," writes the Federal Judicial Center. The Supreme Court will only take on cases that require the level of expertise offered by the Supreme Court. If the court does not feel that the case is worthy of its time then the case will be demoted to the State courts.