InsaneJournal

Requirements: NEWT level EE for all the following subjects: Charms, Defence Against the Dark Arts, Herbology, Potions, Transfiguration, plus a gruelling round of character tests and background checks to determine fitness for the job.
Length of Training: 3 years. Interviews and testing for new trainees occur in August and training proper begins in early September. Exams are held in July of each year, and those who do not perform adequately will be dismissed.
Training Head: Alastor Moody (god rest your new recruit soul)
Courses:

Concealment and Disguise: Teaches many methods of, as one would expect, concealment and disguise, including (but not limited to): Disillusionment Charms, Glamour Charms, the use of Polyjuice Potion, and even how to adopt a foreign accent. (Taught by Alice Longbottom)

Crime Scene Management: Teaches the proper methods of first sealing off a crime scene and then extracting the necessary evidence from it. (Taught by Gawain Robards)

Dangerous Plants: The Dark Side of Herbology: how to handle dangerous plants when confronted with them.

Dark Magic Detection: Teaches how to recognise Dark Magic, as well as how to handle it when confronted with it. Topics include how to determine if someone has been placed under the Imperius Curse and how to recognise and handle cursed objects. (Taught by Alastor Moody)

Emergency Healing: Aurors don't get anywhere near the training that even Mediwitches and -wizards receive, but they are taught rudimentary Healing skills and learn how to Heal themselves as much as necessary, and how to prevent an injured person from dying before medical assistance actually arrives on a scene. (Taught by Joey Proudfoot)

Interrogation Techniques: Teaches many techniques used to extract information from criminals and non-criminals alike. Special topics include the interrogation of suspected victims of the Imperius Curse and the appropriate use of force during questioning. (Taught by Samir Williamson)

Magical Combat: Otherwise known as 'duelling,' teaches advanced methods of duelling, highly complex offensive and defensive spells, counter-jinxes, and tactics and planning in battle, as well as 'Muggle duelling' techniques. Trainees are expected to learn and use non-verbal spells. Magical Combat is physically gruelling and one of the most difficult in Auror training. (Taught by Rufus Scrimgeour)

Potion and Poison Recognition: Teaches how to recognise Dark Potions as well as poisons, as well as counter-potions and antidotes for even the most obscure of brews. Also discusses how to determine whether or not someone is under the influence of a Truth Serum, and when they are beneficial and appropriate to use. (Taught by Kingsley Shacklebolt)

Procedure: Commonly believed to be the most boring course in Auror training, this class teaches the proper (and legal) methods of apprehending, questioning, and processing criminals, witnesses, and victims. Basically, this is the class that tells Aurors all the rules they have to follow, including the standarised methods of contacting St Mungo's, when it is legal and appropriate to use Veritaserum, and how to actually fill out all that paperwork you have. (Held for John Dawlish)

Stealth and Tracking: Teaches a vast array of advanced methods of tracking criminals, as well as determining the location of wizards unknown. (Taught by Frank Longbottom)

Requirements: NEWT level EE for the following subjects: Charms, Defence Against the Dark arts, and Transfiguration, plus a clean criminal record.
Length of Training: 2 years. Applications are processed and new trainees admitted in August, training begins in September, with exams (and dismissals) in July. Note that, though they cover some of the same material, Hitwizard training is considerably less intense.
Courses:

Crime Scene Management: Teaches the proper methods of first sealing off a crime scene and then extracting the necessary evidence from it.

Dark Magic Detection: Teaches how to recognise Dark Magic, as well as how to handle it when confronted with it. Topics include how to determine if someone has been placed under the Imperius Curse and how to recognise and handle cursed objects.

Magical Combat: Otherwise known as 'duelling,' teaches advanced methods of duelling, offensive and defensive spells, counter-jinxes, and tactics and planning in battle, as well as 'Muggle duelling' techniques. Trainees are expected to learn and use non-verbal spells. The Hitwizard Magical Combat course is still physically taxing and difficult, but does not delve into as much detail or as complex spells as Auror training does. For instance, Hitwizards are not trained in use of the Unforgiveables.

Procedure: Commonly believed to be the most boring course in Hitwizard training, this class teaches the proper (and legal) methods of apprehending, questioning, and processing criminals, witnesses, and victims. Basically, this is the class that tells Aurors all the rules they have to follow, including the standarised methods of contacting St Mungo's, and how to actually fill out all that paperwork you have. Hitwizards are also not authorised to use Veritaserum.

Working with Mediwizards: As Hitwizards are often the first on the scene, they have a closer relationship with Mediwizards than Aurors do, and an entire course dedicated to Healing techniques and procedures, including guest lectures by many St Mungo's instructors.

When to Call an Auror: Or, as Hitwizard trainees often call it, "Let's Remind You You Aren't As Good As Them." This course details a myriad of situations that fall out of Hitwizards' hands, and which crimes fall in which department's jurisdiction.

Requirements: The Department of Mysteries has no stringent NEWT-level requirements and will take any minds they see as exceptional, but instead requires passing OWLs in Ancient Runes, Arithmancy, Astronomy, Charms, Defence Against the Dark Arts, Divination, Potions, and Transfiguration. NEWT-level courses chosen will usually determine which chamber an Unspeakable is ultimately assigned to. A series of stringent character tests are required, mainly focusing on the ability to keep a secret. New trainees are also required to sign a waiver agreeing that, should they ever leave the department for any reason, they will be Obliviated.

Length of Training: There are no courses involved in becoming an Unspeakable, but apprentices will spend six months in each chamber (Thought Chamber, Space Chamber, Death Chamber, Time Chamber, Love Chamber, and the Hall of Prophecy) before they are assigned to their permanent location.

Requirements: None -- Wizengamot members are appointed by the Minister for Magic with the assistance of his undersecretaries.

Appointment Procedure: All Heads of Department, the Minister for Magic, and the Senior Undersecretary to the Minister are automatically given a seat on the Wizengamot. The rest of the forty-odd members are appointed by the Minister himself. Generally, each Wizengamot member has a series of undersecretaries and assistants who compromise the readiest pool of Wizengamot appointees, as they are already familiar with the legislative body and its procedures. However, any British wizard who has distinguished himself in the Minister's eyes may be appointed, though those not previously involved in the Wizengamot are much fewer. Department Heads, the Minister, and his Senior Undersecretary all resign their positions when they leave office; other appointees generally stay on for life, though they may be removed with just cause. The Chief Warlock of the Wizengamot is elected by the Wizengamot itself and may also be removed through a Wizengamot ruling.

Duties: The Wizengamot is responsible for both the legislative and judiciary proceedings in wizarding Britain. It passes laws and conducts trials of criminals. As many members of the Wizengamot may have other duties, Wizengamot members may choose how deeply involved they are in the day-to-day running of the body. For major trials and major pieces of legislation, the Chief Warlock of the Wizengamot may call all members to their seats, but some Wizengamot members are seen more frequently than others.