The 1803 ruling in Marbury v. Madison affected the balance of power in federal governments by giving the judiciary a way to check the power of Congress by establishing the power to judicial review, that is to say, the power to review laws and actions from Congress and the President to determine whether they are constitutional.
Weeks before Thomas Jefferson defeated John Adams in the 1800 presidential election, President Adams, along with the Congress, had enacted the Judiciary Act 1801, which created new courts and judges. Adam's intention was to preserve his party’s control of the judiciary.
William Marbury, a Federalist Party leader from Maryland, didn't receive his commission before Jefferson became president and once Jefferson entered the White House, he ordered his secretary of state, James Madison, to withhold the commission. Marbury presented in the Supreme Court petitioning to compel the new Secretary of State to deliver the commision via writ of mandamu, as enabled in the Judiciary Act of 1789.
In the case, it was decided that Madison’s refusal to deliver the commission was illegal, however, it was also established that the provision of the Judiciary Act of 1789 enabling Marbury to bring his claim to the Supreme Court was itself unconstitutional: It gave the Supreme Court jurisdiction over types of cases like Marbury's when Article III, Section 2 of the Constitution didn't allow it. Therefore, the Court had no jurisdiction over the case and couldn't issue the writ that Marbury had requested.
It was the first time in American history that the Supreme Court declared an act of Congress unconstitutional and by this action, it established the judicial branch's power of judicial review.