William Marbury's complain was that he was not appointed to the judge position given to him by President John Adams. After Adams lost his re-election campaign, he attempted to appoint several judges to ensure that federalists maintained power in the federal government. However, the commissions were not delivered before the end of Adam's presidency.
When Jefferson took over the presidency, he refused to give these commissions to judges like Marbury. This is how the controversy began. This ultimately lead to the Supreme Court case Marbury vs. Madison, which resulted in the establishment of judicial review.
