Let us try to simplify what is really a very poignant moment in the political history of the United States.
The importance of the case
Marbury vs. Madison is the pivotal case which precluded the separation of the Judicial branch from the Legislative and the Executive branches. Moreover, this case is the first one which questioned the extent of the power of the President of the United States, as well as the influence of one branch over another.
As we know, all three branches were created to regulate each other and not one overpowers the other. It was not this way back in 1801, and this created a lot of double-standards and very shadowy situations. Moreover, the role of the "President" was still taking shape. There was nothing to "control" a potentially despotic action from the President, especially if the legislative branch supported him.
However, the drama of 1801 changed all this.
The Historical background
The situation was no different than in a change of government in the XXI century: A President is voted out of the White House and, right before ending his presidency, the outgoing President wants to make some last minute changes.
In this case, the outgoing President was John Adams. Worried about losing slots in the Judiciary branch from his own party, he nominated 42 judges from "his team" sort of at a last minute prior to parting the presidency, so that his party's majority would always out rule the incoming President, Thomas Jefferson.
Marbury was one of those judges appointed almost in a shotgun way (they were deemed "The Midnight Judges") and basically accepted his appointment overnight. However, things were about to change.
First, the commissions took too long to process and they were still incomplete by the time that Adams officially left. Therefore, Marbury never really received it. He was merely appointed as a judge, not yet commissioned.
Second, with Jefferson already in power, he cut the judges' positions again, from 42 to 30. Madbury was part of the 12-man cut and he simply turned in a grievance in which he asked Madison (the new Secretary of State) to give him what belongs duly to him.
What makes the case interesting is that, rather than simply denying Marbury anything "because it was done before the new President", there was a proper due process conducted in which his concerns were clearly stated, and they were also answered.
"The" three questions
The three basic questions were a) Whether Marbury was qualified for the job he was appointed to do, b) whether the format of the judiciary committee was the proper forum to process the cause and c) whether Madison actually has the right to ask for Marbury's commission back.
The conclusions were: a) Marbury was qualified; b) He used the proper forum to address his concern...but:
Marbury was sent to the appellate court because Congress cannot give the Supreme Court any mandate to make Madison do anything.
Therefore, Marbury vs. Madison is the case that may have very well separated the three branches, for the first time, and not in favor of the President, but in favor of the constitutional rights of a citizen.
We know however that the outcome was meant to appease everyone: Marbury was rich and unlikely to bother appealing-and they knew it. But the government also wins: an additional force comes into play; three branches, all separate, and all equal will set the records straight from this moment on.