It is structured like a pyramid. At the top of the pyramid is Cornelius Fudge, the Minister of Magic. Fudge is paranoid and believes that Albus Dumbledore, headmaster of Hogwarts School of Witchcraft and Wizardry, is a threat to his authority and therefore constantly tries to have him discredited. He also refuses to believe, in the fourth and through most of the fifth book, that Lord Voldemort has returned, which is an impediment to the work of the Order of the Phoenix.
Below Fudge are various undersecretaries (most notably Dolores Umbridge and Percy Weasley) who are mostly just yes-people who scamper to do Fudge's bidding. Fudge has been known to place them in high positions in organizations furthest from his reach (such as Hogwarts) to gain control over those organizations.
Below them are the heads of the seven departments. These departments are the equivalent of cabinet ministries, such as the U.S.'s State Department and Department of Justice.
Department of Magical Law Enforcement
Arguably the most important department, this department is a combination of police and justice facilities. It includes the headquarters of the Aurors, a team of experienced wizards that searches for Death Eaters. It also includes the Improper Use of Magic Office which punishes wizards for using magic in inappropriate ways, at the wrong time, or in violation of a magical law. Finally, there is the Wizengamot, a sort of wizarding supreme court which includes Fudge and his cronies. This department was once headed by Barty Crouch, but his son, a death eater, killed him. The current Head of Department is Amelia Bones.
Department of Magical Games and Sports
The most relaxed department deals with such sports-related pastimes as organizing the Quidditch World Cup. Ludo Bagman used to be the Head of Department here, but his gambling problem forced him to flee.
Department of Magical Accidents and Catastrophes
If one of your spells has gone awry or you accidentally performed magic in front of a muggle, you can count on this department to send a squad which will reverse magic or wipe memories clean.
Department of Magical Transport
Controls the floo powder network, and hosts the magical equivalent of the DMV, which grants licenses to witches and wizards so that they can apparate, or teleport.
Department for the Regulation and Control of Magical Creatures
Divides animals into beings (intelligent life), beasts (not intelligent), and spirits, and liaises with goblins and centaurs.
The department is extremely mysterious, as its name implies, and is totally unknown to all wizards except those few, known as Unspeakables, who work there.
Sixteen years before Harry Potter and the Order of the Phoenix, Sybill Trelawney made a prediction about Lord Voldemort and Harry Potter. A Death Eater overheard this prediction, but only part of it. Voldemort decided to kill Harry Potter, believing he was fulfilling the prophecy: instead, he lost his powers.
After his powers were restored in Harry Potter and the Goblet of Fire, Voldemort decided he had to get his hands on Sybill's prophecy, now stored in the Department of Mysteries. The trouble was that only he or Harry Potter could take them from the Department of Mysteries. As Lord Voldemort wasn't about to walk into the Ministry of Magic, he decided to lure Harry there.
This plan nearly succeeded, but the prophecy was destroyed before Voldemort could obtain it. There was a struggle in the Department and Sirius Black was killed. When Cornelius Fudge saw Voldemort at the Ministry of Magic, he was forced to acknowledge that he was wrong in denying that Voldemort had indeed returned.
Finally, there are many of minor offices, such as the Misuse of Muggle Artifacts Office where Arthur Weasley works.Department of International Magical Cooperation
The British International Confederation of Wizards is based here, as are offices that regulate international magical law. This is where Percy Weasley began to climb the ladder of success, and where Barty Crouch spent his final days in the Ministry.Department of Mysteries
This department works to record prophesies, and uncover the secrets of death and time. Prophesies are magically protected, so that the only people who can touch them are the subject of the prophesies. All others are afflicted with instant madness. Given these magical protections, it is unclear why the Department bothers to keep the prophesies.