The United States Supreme Court is a federal court, meaning in part that it can hear cases prosecuted by the U.S. government. (The Court also decides civil cases.) The Court can also hear just about any kind of state-court case, as long as it involves federal law, including the Constitution. And any case can involve federal law. For example, a defendant's challenge to the basis for a police search implicates the Fourth Amendment, and is therefore within the Supreme Court's jurisdiction.
The Supreme Court gets to choose the appellate cases it will consider—it grants certiorari (or "cert") when it agrees to decide a case. Grants of cert are quite rare—the Court typically agrees to take on somewhere between 100 and 150 of the 7,000 cases it comes across each year. (See "About the Supreme Court.")
For further reading on a related topic see State vs. Federal Prosecution.