This is John Adams, he is leaving office tomorrow and is just finishing a few things up first. He had appointed some judges in the Federalist Party to the District of Columbia to get jobs there. Everything was set in place for them to start working, the only thing left was mailing out commission letters. He didn't have enough time to do that but hoped that Jefferson would.
Don't send out those commissions James!
Meet Thomas Jefferson, the new president. His secretary, James Madison was supposed to send out those commissions. Jefferson didn't want him to though and told him not to.
It's been months and I still haven't received my commission! That's it, I'm suing!
This is Marbury. Marbury is one of the judges that was supposed to get sent a commission that allows him to work. Since James Madison never delivered it he's taking the case to the Supreme Court.
The court takes on the case. At first, they struggle with coming to an answer, but then, they realize that because of Article III of the Constitution they can't force Madison to give out commission. The case goes to Madison, it's a unanimous 4-0.