The plaintiff in this case was a man who had been given a judicial appointment in the dying days of the Adams Administration. However, the appointment letter was never delivered and the new administration of President Thomas Jefferson refused to honor Marbury's appointment. Marbury took James Madison, the secretary of state, to the Supreme Court in a bid to try to obtain the judicial commission which he believed was rightfully his.
In a landmark judgment, the Supreme Court under Chief Justice Marshall ruled that Marbury was not entitled to his appointment. This was largely a political decision, as the Court was made up of Federalists, who didn't want to give the new Democratic-Republican Administration and Congress an excuse to reduce the Supreme Court's powers.
The Court ruled that the law that had given it jurisdiction in the Marbury case was invalid. In other words, the Court struck down a law that had given it more power. In doing so, however, the Court paradoxically increased its power by unilaterally exercising the right of judicial review, that is the power to strike down legislation as unconstitutional, and that power has remained with the Supreme Court ever since, making Marbury v Madison one of the most important cases in American legal history.