What does the judicial branch of the federal government do?

The judicial branch of the federal government is made up of the Supreme Court and other appellate courts throughout America.  The Supreme Court was created by Article III of the U.S. Constitution.  It handles whether or not a law is constitutional through a practice called judicial review.  The precedent of this was created with the case of Marbury v. Madison (1789) in which Jefferson's Secretary of State James Madison sued to keep John Marbury off the federal court bench.  Marbury was part of John Adams's "midnight justices," a group appointed in the last days of the Adams administration to keep Federalist laws in place.  It was really Supreme Court Justice John Marshall, who would go on to serve over thirty years past this ruling, who created the process of judicial review.  Cases are argued in appellate courts and they can be appealed all the way to the Supreme Court.  Supreme Court justices are appointed by the President with Senate approval, and the appointment is a lifetime position unless the judge voluntarily retires.
Since that time, the Supreme Court has been part of famous if not controversial hearings.  They ruled in favor of slaveowners in the case of Scott v. Samford, when Chief Justice Taney ruled that slaves could not sue in court.  They also ruled in favor of integration in the case Brown v. Board, which ended segregation in schools and was used as a precedent to promote integration throughout America.