During the last days of Adams' government, the Congress established 42 judges of peace for the District of Columbia.The Senate confirmed the appointments, the president signed them but the outgoing secretary of state did not deliver the appointment to four justices of the peace, including William Marbury. On the other side, James Madison, the new seccretary of state under President Jefferson's government, refused to deliver the appointment records because the new government was irritated by the outgoing government maneuver to try control the judiciary by appointing members of its party just before ceasing in government. However, Marbury appealed to the Supreme Court to order Madison to deliver his record.
The mayority decision of the court was that Malbury had a right to his commision but that the Judiciary Act of 1789 (that established the federal judicity of the United States) was unconstitutional, then Congress could not give the Supreme Court the power to issue an order granting Marbury his commission. This case established a very important precedent (legal decision used as an example in later court cases).
It was fundamental to the workings of the supreme court because after it the U.S. Supreme Court was able to established the principle of judicial review in the United States, meaning that American courts have the power to strike down laws, statutes, and some government actions that contravene the U.S. Constitution.