One of the first acts was to repeal the Judiciary Act of 1801. This created a number of new judge ships. President Adams had attempted to fill the vacancies. He sent out 17 commissions to James Madison, but the commissions had not been delivered.
Marbury sued the secretary of state , James Madison, and forced him to deliver his commission as a justice of peace.
Marshall understood that if Marbury was awarded this, the Jefferson Administration would ignore it. Marshall declared that Madison should have delivered the commission to Marbury.
The chief justice ruled that the court could not grant the writ because section 13 of the Judicial Administration of 1789, which granted to do so, was unconstitutional.
Marbury was denied his commission which pleased Jefferson.
The Supreme Court would not declare another act of congress unconstitutional. Judicial review, the supreme court is the final interpreter of all laws.