Departments in the LAPD are different parts of the police force. Each one investigates different kinds of crimes that plague the city of Los Angeles and have different brief rooms across the various LAPD police stations. Cole Phelps will be a part of a few different departments and will solve several cases for each one of them. While doing so, he will have the aid of a different partner for each department and also a different Captain that administers the department he is working at.
The Patrol Department is a section of the LAPD concerned with monitoring assigned geographical areas in Los Angeles. This desk is mainly a tutorial desk, and if you skip tutorial cases, you will then start the game at the Traffic Desk. When Cole Phelps joins the LAPD, he starts off in this department as a beat cop at the Wilshire Police Station, but the department quickly notes that he has the ability to investigate crimes, and promotes him to Detective. Cole's partner in this department is Ralph Dunn
The Traffic Department is a section of the LAPD concerned with hit-and-runs, felony driving, and car burglaries. The Traffic crime desk also investigates vehicular manslaughter and murder cases, and collision reports involving serious injuries or fatalities where a car may be involved.
Cole Phelps starts in the Traffic Department at the Central Police Station when he first becomes a detective in the LAPD. Cole's partner in Traffic is Stefan Bekowsky. The Captain of the department is Gordon Leary.
The Homicide Department is a section of the LAPD concerned with the investigation of murders within the city of Los Angeles. Cole's partner in Homicide is Rusty Galloway. The Captain of the department is James Donnelly.
The Administrative (Ad) Vice Department, also known as Vice Narcotics is a section of the LAPD concerned with offenses involving prostitution, drugs, lewdness, lasciviousness and obscenity. Cole is transferred to Vice at the Hollywood Police Station and is partnered with Roy Earle In Vice. The Captain of the department is Captain Lazarus Cafarelli.
The Arson Department is a section of the LAPD concerned with maliciously, voluntarily, and willfully setting fire into buildings, or other property of another or of burning one's own property for an improper purpose, as to collect insurance. When Cole returns to Wilshire Police Station to work in Arson, his partner is Herschel Biggs. The Captain of the department is Lachlan McKelty.
Two crime desks were not included the final version of L.A. Noire: Bunco (Fraud) and Burglary (Robbery). Team Bondi's Brendan McNamara explained in an interview that Bunco and Burglary were cut from the game early on: "We had so much to work with that some material never made it past the conceptual stage: we never ended up shooting capture or creating substantial code for Burglary and Bunco, for instance, although they were part of the original idea."  The names of some of the tracks on the soundtrack also reference the Bunco and Burglary desks.
Bunco, also known as the Fraud Division of the LAPD is responsible for investigating fraud and other white-collar crimes. Its department is located at Wilshire Police Station. It is possible that this desk, like Burglary, took place during the six months between Traffic and Homicide.
The Burglary Department of the LAPD investigates robberies involving private homes, home invasions and thefts. When Cole is introduced to the Homicide desk, Captain Donnelly mentions he was transferred from Burglary. It is assumed that this desk took place during the six months between Cole's service in Traffic and his promotion to Homicide. Harold Caldwell may have been Cole's Partner in this desk.
Desks Not Featured
The Robbery Department is a desk of the LAPD responsible for investigating thefts, such as store robberies, break-ins, etc. It is possible that Robbery is a subdivision of the Burglary Desk.
The Auto Theft Department of the LAPD is responsible for investigating thefts of vehicles and other such crimes. It is possible that Auto Theft is part of the Traffic Desk.
Forgery Division is a Department of the LAPD responsible for investigating forgeries used in crimes. It is possible that Forgery is a subdivision of the Bunco Division.