The Gist: Before being replaced by the newly elected president, the outgoing president appointed new judges. Out of spite, the newly elected president effectively prevented some of the newly appointed judges from taking the bench by refusing to formally finalize their appointments. One of the people being prevented from taking the bench brought his case against the executive branch directly to the Supreme Court, asking the court to force the president to finalize his appointment.
The Supreme Court decided that though the president was required to finalize the appointment, the court could not force him to do it because it did not have the authority (the jurisdiction) to do so. Most importantly, it is in this case that the Supreme Court established the concept of judicial review, or the court's duty to "say what the law is."
The Details: In the presidential election of 1800, Thomas Jefferson defeated John Adams. Though the election was decided in February of 1801, Jefferson and newly elected Congress members would not take office until March, giving Adams (and a Congress with a majority that favored Adams' policies) time to execute a few more moves before being replaced by Jefferson (and a Congress with a majority that favored Jefferson's policies). Congress took advantage of this time by passing an act that expanded the Federal court system, and Adams took advantage of this time by appointing judges and justices of the peace to fill the vacancies that had been recently created by the new act.
For judicial appointments by the president to be valid, however, the commissions had to be delivered to the newly appointed judges/justices of the peace. When Jefferson took office, there were still a few commissions of judges appointed by Adams that had not been delivered. Jefferson refused to deliver the remaining commissions. William Marbury was one of the newly appointed justices of the peace who had been slighted by Jefferson's actions, so he brought his case directly to the Supreme Court, demanding that the court order the newly elected president to deliver the commission to him.
The court recognized Jefferson's actions were a violation of Marbury's legal rights and that the president had a legal obligation to deliver the commission. However, the court also stated it could not force the president to do so because, under Article III, Section 2, Clause 2 of the Constitution, it did not have original jurisdiction to hear this type of case--it only had appellate jurisdiction. In other words, instead of bringing the case directly to the Supreme Court as he had done (original jurisdiction), Marbury was required to take his case to the lower courts first and appeal to the Supreme Court (appellate jurisdiction) if he didn't agree with the lower courts' decision. Under the Constitution, the Supreme Court has original jurisdiction under very limited circumstances. Though debatable, the court determined it did not have original jurisdiction based on its interpretation of the Constitution.
Most substantially, however, the court stated that it is the duty of the Judicial Branch to determine what the law is and whether laws passed or executed by the other two branches are constitutional. This was the first formal establishment by the United States Supreme Court of its authority to exercise judicial review over the government's actions, as the concept of judicial review is not explicitly stated in the Constitution. The court made a bold move by unilaterally asserting its power to check and balance the other two branches of the government, and it paid off--though many of the Supreme Court's decisions have been seriously questioned and examined, the Supreme Court's overall authority to determine what the law is and interpret the constitutionality of the government's actions has never been jeopardized.