Let me see if I can give you a high-level historical analysis:

1. John Adams and Thomas Jefferson had extremely different visions of the federal government and its role in the American experiment. Adams came to be associated with the Federalist Party, and Jefferson with the Democratic-Republicans.

2. Adams lost his re-election race to Jefferson. In his waning days in office, President Adams sends out a bunch of commissions, that is, appointments to federal office, hoping to essentially pack the bureaucracy with Federalists. Adams hoped that his last-minute appointees will stymie the new Jefferson presidency at every turn.

3. Some Adams appointments went out on literally his last day in office -- up to the stroke of midnight. These so-called "midnight appointees" had... questionable legitimacy, but none so questionable as those whose appointments were delivered after midnight, or signed by Adams but not delivered at all.

4. William Marbury was one such appointee. Adams wrote him an appointment to a Justice of the Peace position in the District of Columbia. However, the appointment was never delivered.

5. Marbury sued James Madison, the new Secretary of State, asking him to deliver his commission. Here is a critical part: Marbury filed his suit directly in the Supreme Court. This is not normal. The Supreme Court has very little "original jurisdiction": that is, it has very little opportunity to take a first crack at a lawsuit. Most of its jurisdiction is "appellate": meaning, you have to lose somewhere else, and then ask the Supreme Court to step in.

6. However, the Judiciary Act of 1789 gave the Supreme Court "original jurisdiction" over petitions for a writ of mandamus against a government official. A writ of mandamus is basically an order directing that a government official perform some ministerial act. So Marbury asked for mandamus relief against Madison pursuant to the Judiciary Act of 1789.

Now let's get into law.

1. This teed up a problem for John Marshall, Chief Justice of the Supreme Court. If Marbury won, it would be viewed as a huge political victory for ex-president Adams, and a stunning early rebuke for Jefferson's administration. If Marbury lost, it would be viewed as a poke in the eye to the Federalist Party, which while out of power, would surely return some day soon.

2. So Marbury split the baby. How?

3. He ruled ("held" in legal parlance) that Marbury was entitled to his commission. Ordinarily, then, Marbury would be entitled to his writ of mandamus.

4. BUT he also "held" that the Judiciary Act had expanded the Supreme Court's "original jurisdiction" beyond the limits allowed by the Constitution. The Constitution did not permit Congress to vest the Supreme Court with original jurisdiction over equitable writs like mandamus.

5. So, and here is the big point: Marbury had a right to his commission but, since the Judiciary Act was unconstitutional, he had no remedy in the Supreme Court. Marbury would have to find the right court and sue in it.

Now here's an interesting point. Marbury stands for two propositions today:

• The Supreme Court may invalidate a law of the state or federal governments if it violates the Constitution, because the Constitution is the supreme law of the land, and "it is emphatically the province and duty of the judicial department to say what the law is"

• Every constitutional right must have a remedy somewhere in the judicial system. But a litigant must use the correct remedy to secure his right. (That's why Marbury lost.)

Interestingly, though, Marbury might not have had a remedy anywhere in the court system. It's unclear whether any federal court had jurisdiction in 1803 to issue an original writ of mandamus. So the great irony of Marbury is it's the bedrock of our judicial system here, but one of its two great holdings is a farce!!

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TLDR: Marbury v. Madison arose out of a huge political fight between John Adams's wing of the founding generation, and Thomas Jefferson's wing. Because Chief Justice John Marshall was a very clever fellow, he resolved the case in a compromise that left both parties angry, but had the side effect of clearly defining the Supreme Court's authority of judicial review. Today we remember the case for its side effect.

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Sources:

• Gordon Wood, Empire of Liberty

• Akhil Amar, America's Constitution: A Biography

• Hart & Wechsler, Federal Courts & the Federal System

• My constitutional law textbook, which I don't have on hand right now; I'm going from my notes.

• Countless lectures by Professor Burt Neuborne, who's rightly fascinated by this case.

The Constitution is the supreme law in the United States. No law passed by the Congress or state legislatures must conflict with it, and federal laws must generally have some source in the Constitution.

Marbury was appointed justice of the peace by the Adams presidency, but did not receive his signed and sealed commission as Secretary of State James Madison declined to deliver it (they were of different parties). Marbury sued under the Judiciary Act, which purported to give the Supreme Court original, as opposed to appellate, jurisdiction over such cases.

Chief Justice Marshall wrote that the original jurisdiction of the Supreme Court as outlined in the Constitution was exclusive and could not be modified by Congress, and thus the Judiciary Act was unconstitutional. He wrote that when Acts of Congress conflict with the Constitution, such acts are not laws at all, and that the federal courts are bound to interpret the Constitution, with the Supreme Court being the final say on the issue. This is the concept of judicial review. Courts in ordinary practice of a case decide what happens when two laws conflict with each other, and as the Constitution is, by its own writing, the supreme law of the land, it controls in every case of conflict with another law.

The decision was criticized both when it was made and subsequently, both for Marshall's jurisprudence in deciding the case and for its practical effect on the national constitutional system. It vastly increased the power of the Supreme Court from how it originally operated. John Jay, the first Chief Justice, resigned from the Court to run for Governor of New York. Imagine that happening today.

In short, the case is hugely important: it is the source of what is now the Supreme Court's most sacred role: being the final arbiter on the constitutionality of laws.

Of course, there were several instances where the President has stated he has the right to determine what is constitutional, most notably President Lincoln during the Civil War. This is called departmentalism, and is mostly dead today. Also, it's a story for another question.