Answer: The origin of the case was somewhat trivial, but had great implications for the role of the Supreme Court in government. Marbury was appointed by John Adams, the president before Madison, as a district judge in Washington DC. When Madison became president, he didn't deliver the papers to finalize Marbury's appointment.
Marbury took him to Court, and although the Court initially sided with Marbury, the court, with John Marshall serving as Chief Justice, ultimately determined that the law that allowed Marbury to take the case to court was not constitutional. This meant that the law was struck down.
This was the first incidence of the Supreme Court exercising judicial review, the review of laws to determine constitutionality and their rejection if they are not, in the history of the United States. It was a landmark case not for the spat between Marbury and Madison over a district judgeship, but because it marked a huge expansion of the power of the Supreme Court (and thus the judicial branch).
We have seen the power of judicial review exercised in many cases since this one, such as Miranda vs Arizona (which established the law that police must read you your 'Miranda Rights' when they arrest you) and Plessy vs Ferguson, which determined that laws governing "seperate but equal" facilities for people of different races were in theory inherently unequal, and in practice clearly offered worse facilities to people of color.
