Minister of Magic
Departments, committees, etc.
Rules and Regulations
Personnel
International Wizardry
Other organizations

The Ministry of Magic is the governmental authority of the Wizarding World in Britain. The basic mission of the Ministry is to keep the Muggles from finding out about an entire culture of magical folk. It's a huge, complicated, and inefficient bureaucracy, just like most governmental organizations.

The Ministry consists of seven departments (FB) of which the Department of Magical Law Enforcement is the largest. All of the other departments are in some ways answerable to it, with the possible exception of the Department Of Mysteries (FB). 

The Ministry has its headquarters, or at least some of its offices, in London (PA11). It consists of numerous Departments, Committees, Boards, and Offices whose job it is to make and enforce laws and regulations. Besides keeping the activities of Witches and Wizards hidden from the Muggles (and swooping in with Memory Charms when things get out of hand), the Ministry also handles commerce and international relations.

There are a number of law enforcement agencies within the Ministry. They maintain order, enforce laws and regulations, deal with breaches of security, and operate Azkaban, the Wizard's Prison. This high-security fortress is located on a tiny island far out to sea. It is guarded by Dementors. (PA) The Ministry also maintains squads of Hit Wizards to deal with criminals and elite Aurors to track down Dark Wizards.

The Muggle Prime Minister is aware of the Wizarding World. Presumably this is also true of the President of the United States and of other world leaders. When Sirius Black escaped from Azkaban, Cornelius Fudge informed the Prime Minister the escapee was a dangerous wizard because he posed a threat to Muggles as well as Wizards (PA). In extreme cases where a magical creature has been seen by Muggles and the usual interventions won't suffice, the Office of Misinformation (part of the Department for the Regulation and Control of Magical Creatures) works with the Muggle Prime Minister to concoct a believable lie to give to the public (FB).

Other countries have similar governing bodies; Bulgaria, for example, has it's own Ministry of Magic, the Minister of which, Mr. Oblansk, sat in the Top Box to watch the World Quidditch Final in 1994 (GF8). There is also a Ministry of Magic in Norway (QA) and other countries. All of these governmental organizations have signed the International Statute of Wizarding Secrecy of 1692 which set down many requirements for individual countries to keep the Muggles unaware of the Wizarding World. This requires the Ministries of each country to monitor activities such as Quidditch and to protect and hide the various magical beasts which live within their borders (FB, QA).

The Wizards' Council was the predecessor to the Ministry of Magic. Barberus Bragge was the Chief of the Council in 1269. Burdock Muldoon was Chief of the Wizards' Council in the 14th Century (FB, QA). He was followed in office by Elfrida Clagg, who is generally regarded as being more enlightened than her predecessors (FB). Elfrinda outlawed the hunting of the Golden Snidget, banned its use in Quidditch, and set up a reservation to protect the tiny bird (QA). 

Minister of Magic
Cornelius Fudge  (c.1990's)
previous Minister of Magic: Grogan Stump (c. 1811) (FB)
 

Departments, Committees, etc:

Although no organizational chart is available for the Ministry, I have tried to organize this list by Departments. Of course, this is all speculation. The Ministry is a convoluted bureaucracy with plenty  of overlap and redundancy, which makes it hard to figure out where each part belongs. We do know that there are seven Departments in all (FB), with the Department of Magical Law Enforcement being the largest, but there is mention of eight departments in the books and no hint of which one might actually NOT be a Department at all, but a subdivision thereof just pretending. The precise organizational structure below, then, should not be considered canon,  but at least it gives this list some logic. I have also listed witches and wizards who are known to work in each department.

Department of Magical Games and Sports
Head: Ludo Bagman
Other personnel: Bertha Jorkins (GF3)
Previous Head: Hamish MacFarlan (after 1968)

• Organized the Quidditch World Cup final and also the Triwizard Tournament
•  this Department was formed in 1750 (QA6)
•  responsible for hiding Quidditch from Muggle eyes (QA)

• deals with major crimes and disasters
• this "Department" seems like a likely candidate to be part of another Department, probably the Department of Accidental Magic Reversal

Improper Use of Magic Office
Personnel: Mafalda Hopkirk (CS2)

• sent Harry a warning letter when they detected magic at the Dursley's house

• keeps enchanted items out of the hands of Muggles
• participates in raids to keep Dark Magic items out of circulation

Department of International Magical Cooperation
Head: Barty Crouch Sr.
Other personnel: Percy Weasley (GF3)

• works with magical governments of other countries
• sets standards for trade
• creates regulations for things like cauldron thickness
• worked with Dept. of Games on the Triwizard Tournament

Magical Law Enforcement Squad 

• employs Hit Wizards to handle dangerous criminals such as Sirius Black (PA11)

• Dark Wizard Catchers
• Alastor Moody, Frank Longbottom

Something of a tribunal or court of law, held in a dungeon; the accused is magically bound to a chair in the center of the room. A jury sits on the right of the prosecutor who states the case against the accused and eventually asks for a verdict. This court may possibly be located at Azkaban, since prisoners are escorted and guarded by groups of Dementors (GF30).

•  handled the complicated problems of transporting folks to the World Cup
•  set up a network of Portkeys
•  worked out where it was safe to Apparate into the area without being seen

• regulates which fireplaces are connected to the Floo Network

• navigate traffic with magical ease (PA5)

• handles complaints about magical creatures, including goblins and ghosts
• regulates and monitors breeding of new magical creatures (GF24)

Beast Division
Newt Scamander, after 1917 (FB)

Committee for the Disposal of Dangerous Creatures
Walden Macnair
another aging wizard (PA11)

• ordered Buckbeak destroyed after he injured Draco Malfoy
• office is in London

Office of House-Elf Relocation
Newt Scamander, c.1915-17) (FB)

Werewolf Registry (FB)

Werewolf Capture Unit (FB)

Centaur Liaison Office (FB)

• no Centaur has ever actually used this service

Goblin Liaison Office
Cuthbert Mockridge (GF7)

Werewolf Support Services (FB)

Spirit Division (FB)

• Probably the ones who handled Olive Hornby's complaint about Myrtle haunting her; Myrtle was forced to return to Hogwarts and haunt the place of her death: a bathroom. (GF25)
• Formed when Ghosts argued that they weren't Beings, they were clearly "Has-Beens" and deserved special treatment (FB)

Rules and regulations:

• Decree for the Reasonable Restriction of Underage Sorcery (CS2, CS5, PA3)
• Muggle Protection Act (CS4) written in part by Arthur Weasley
• Guidelines for the Treatment of Non-Wizard Part-Humans (GF10)
• Werewolf Code of Conduct (1637) (SS16)
• Werewolf Register (1947) (FB)
• Registry of Proscribed Charmable Objects (GF7)
• International Code of Wizarding Secrecy (GF26)
• International Ban on Dueling (GF23)
• All Animagi must be registered with the Minstry of Magic (PA, GF)
• Ban on Experimental Breeding (1965) (FB) written by Newt Scamander
• Code of Wand Use (GF9) clause 3: no non-human creature permitted a wand
• there are three Unforgivable Curses (life sentence in Azkaban) (GF14)
• Killing Curse - ("Avada Kedavra") - unblockable, kills instantly
• Cruciatus Curse - ("Crucio") - inflicts intense pain and suffering
• Imperius Curse ("Imperio") - total control over victim's actions
• There is legislation about what you can conjure and what you can't (SN)
• The use of Veritaserum is closely controlled by Ministry guidelines (GF27)
• You have to have a license to Apparate (Sch2, GF)
• Trade Restrictions:
• Class A Non-Tradeable Goods 
• (e.g. dragon eggs) (FB)
• Class B Tradeable Materials (Dangerous and Subject to Strict Control) 
• (e.g. Erumpent horn) (FB)
• Ban on Importing Flying Carpets (GF

• Dark Force Defense League (CS6, GF31)
• May or may not be directly affiliated with the Ministry of Magic
• Order of the Phoenix (OP)
• We'll have to wait for book 5 to find out for sure, but my guess is that this is the "Old Crowd" that Dumbledore refers to at the end of GF, all strongly loyal to Dumbledore and ready to fight against the rising of the Dark Lord.

International Wizarding

• International Confederation of Wizards (SS4, GF17)
• International Federation of Warlocks (PA3)
• Ministries of Magic: 
• Bulgarian (GF8)
• Andorran (GF28)
• Norwegian (QA)
• German (FB)
• Burkina Faso (FB)
• a subcommittee of Sardinian sorcerers (CS9)
• Transylvanians are to sign the International Ban on Dueling (GF23)