What is the decision of the court in Marbury v. Madison, and what was the rationale behind it?

In the landmark case of Marbury v Madison (1803), the US Supreme Court arrogated to itself the right of judicial review. And ever since, the Court has reserved that right to invalidate laws and administrative decisions which it deems unconstitutional.
In relation to the specifics of the case, the Marshall Court put forward a carefully-crafted political compromise: in essence, it gave something to both sides in the case, Federalists and Republicans alike. The Federalist plaintiff, Marbury, did not receive the judicial commission to which he believed himself entitled and which brought him to take the Jefferson Administration to court in the person of James Madison, Jefferson's Secretary of State. Chief Justice Marshall knew that if he overturned the Administration's refusal to deliver Marbury's commission appointing him as a judge, then he would inflame an already tense political situation. The potential consequences for the Supreme Court could be damaging in the extreme.
At the same time, by establishing the Court's authority to review the actions of the executive and legislative branches of government, Marshall ensured that Federalists such as himself still retained a significant degree of political power within the system, despite their recent defeat at the election of 1800.