The Department of Magical Law Enforcement is the largest department at the Ministry of Magic. Arguably the most important of the various departments, it is a combination of police and justice facilities, and is roughly the equivalent to the Home Office of Muggle Britain or the Department of Justice in the United States. All other departments are answerable to this one, with the exception of the Department of Mysteries. 
The department is located on the second level of the Ministry of Magic, and includes the following divisions:
An Auror is a highly trained specialist officer who investigate crimes involving the Dark Arts and apprehend Dark Wizards. According to Minerva McGonagall, the Auror Office takes in new recruits with a minimum of 5 N.E.W.T.s (with marks no lower than "Exceeds Expectations"). She suggests that Potions, Defence Against the Dark Arts, Transfiguration, Charms, and Herbology N.E.W.T.s are the most appropriate for someone who aspires to be admitted to the training program. (A potential recruit will also have to pass "...a series of character and aptitude tests.")
Nymphadora Tonks mentions that two of the program's courses of study are "Concealment and Disguise" and "Stealth and Tracking" and that the training is hard to pass with high marks (a requirement). Aurors are the magical equivalents of Muggle counter-terrorism operatives and, during the First Wizarding War, Aurors were authorised to use the Unforgivable Curses on suspected Death Eaters; specifically, they were given the licence to kill, coerce and torture them.
Aurors significant in both the First and Second Wars include Alastor Moody, Nymphadora Tonks, Kingsley Shacklebolt, John Dawlish, Frank and Alice Longbottom, Rufus Scrimgeour, Gawain Robards, Proudfoot, Savage, and Williamson. In addition, Harry Potter joins the department at the age of 17 and is promoted to department head in the year 2007. Ron Weasley becomes a member of the Auror office as well.
Many of the Dark criminals first duel with the Aurors sent to arrest them before finally giving up their freedom. Aurors are also used to protect high-profile targets such as Harry, Hogwarts, and the Muggle Prime Minister.
Department of Intoxicating Substances
Improper Use of Magic Office
This office, the Improper Use of Magic Office, punishes violations of the International Statute of Wizarding Secrecy. This includes underage witches and wizards intentionally using magic, which suggests that this office may be in charge of the Trace, permitting them to monitor the magic used by those under the age of seventeen. Mafalda Hopkirk worked in this office until at least 1998.
Misuse of Muggle Artefacts Office
This office regulates the use of magic on Muggle objects and confiscates those which have been illegally bewitched. One of the laws they enforce is the prohibition on magically altering a Muggle vehicle with the intent to fly it. It was headed by Arthur Weasley until his promotion in 1996.
Office for the Detection and Confiscation of Counterfeit Defensive Spells and Protective Objects
This office was created in 1996 by Minister for Magic, Rufus Scrimgeour, in response to the growing threat of Lord Voldemort and the rise of dangerous or dubious spells and products which claimed to provide protection against the Dark Arts. Arthur Weasley was promoted to head this operation. Its duties involved preventing the trade of fake and/or useless spells, potions and artefacts that appeared during the Second Wizarding War.
Wizengamot Administration Services
An office which undertakes administrative and clerical duties for the Wizengamot. Its function seem analogous to Muggle court registries; this would typically include maintenance of court documents, hearing dates, judges' schedules, and the administration of legal proceedings.
Magical Law Enforcement Patrol
The Magical Law Enforcement Patrol (formerly the Magical Law Enforcement Squad) is tasked with general law enforcement, like with the Muggle police. In the 1920s it was headed by Bob Ogden, who led a team to Little Hangleton to arrest Marvolo and Morfin Gaunt for crimes against both Muggles and the Ministry, since the House of Gaunt had avoided previous attempts at arrest.
Hit Wizards comprise a team of highly trained wizards tasked with arresting dangerous criminals. The entry requirements include five O.W.L.s, one of which must be Defence Against the Dark Arts. The equivalent of a Muggle S.W.A.T. Team. Members have a personal bed reserved at St Mungo's Hospital for Magical Maladies and Injuries. In 1993, Cornelius Fudge suggested that only Hit Wizards should attempt to capture Sirius Black, an Azkaban escapee then believed to be a highly dangerous criminal.
Ministry of Magic Witch Watchers
Ministry of Magic Witch Watchers are a special Ministry force that presumably is stationed at strategical places in search of fugitive criminals.
Administrative Registration Department
The Administrative Registration Department is responsible for the registration of the wizarding population. After the Fall of the Ministry of Magic, the Muggle-Born Registration Commission was created under this department.
- Elphinstone Urquart: senior officer of the department in the 1950s; deceased husband of Minerva McGonagall.
- Justus Pilliwickle: former department head; one of the most notable on the post.
- Barty Crouch Sr.: former department head, demoted to head of the Department of International Magical Cooperation after his son's exposure as a Death Eater in 1981.
- Amelia Bones: former department head, killed by Lord Voldemort in the summer of 1996.
- Pius Thicknesse: former department head, subjugated by Yaxley and promoted to Minister for Magic.
- Yaxley: former department head, during the Death Eaters' control of the Ministry of Magic.
- Hermione Granger: after working with the Department for the Regulation and Control of Magical Creatures, she eventually took a position high up in this department
- Harry Potter: Head of the Auror Office from 2007.
- Harry Potter and the Deathly Hallows (First mentioned)
- Harry Potter and the Deathly Hallows: Part 1 (Mentioned only)
- Fantastic Beasts and Where to Find Them (real) (Mentioned only)
- Daily Prophet Newsletters (Mentioned only)
- Pottermore (Mentioned only)
Notes and references