Ship gun fire-control system
The guns controlled by it were typically 5 inch weapons: the 5-inch/25 or 5-inch/38.

Deployment: destroyers (1 per vessel, total 48) 
8 Farragut-class(launched ca. 1934)
18 Mahan-class(ca. 1935) (Cassin, Downes later rebuilt with Mk37)
4 Gridley-class(ca. 1937)
8 Bagley-class(ca. 1937)
10 Benham-class(ca. 1938)
heavy cruisers
7 New Orleans-class(launched ca. 1933): for the 5"/25 secondary battery
Wichita(1937): for the 5"/38 secondary battery
light cruisers
9 Brooklyn-class(launched ca. 1937): for the 5"/25 and 5"/38 secondary batteries

Mark 34 GFCS: The Mark 34 was used to control the main batteries of large gun ships. Its predecessors include Mk18 (Pensacola class), Mk24 (Northampton class), Mk27 (Portland class) and Mk31 (New Orleans class)

Deployment: 2 Alaska-class large cruisers (2 per vessel)
heavy cruisers
Wichita (2x)
14 Baltimore-class (2 per vessel, 28 total)
several Northampton-class (as upgrade)
Portland-class (as upgrade)
light cruisers
9 Brooklyn-class
27 Cleveland-class

Mark 37 GFCS: According to the US Navy Bureau of Ordnance,

While the defects were not prohibitive and the Mark 33 remained in production until fairly late in World War II, the Bureau started the development of an improved director in 1936, only 2 years after the first installation of a Mark 33. The objective of weight reduction was not met, since the resulting director system actually weighed about 8,000 pounds (3,600 kg) more than the equipment it was slated to replace, but the Gun Director Mark 37 that emerged from the program possessed virtues that more than compensated for its extra weight. Though the gun orders it provided were the same as those of the Mark 33, it supplied them with greater reliability and gave generally improved performance with 5-inch (13 cm) gun batteries, whether they were used for surface or antiaircraft use. Moreover, the stable element and computer, instead of being contained in the director housing were installed below deck where they were less vulnerable to attack and less of a jeopardy to a ship's stability. The design provided for the ultimate addition of radar, which later permitted blind firing with the director. In fact, the Mark 37 system was almost continually improved. By the end of 1945 the equipment had run through 92 modifications—almost twice the total number of directors of that type which were in the fleet on December 7, 1941. Procurement ultimately totalled 841 units, representing an investment of well over $148,000,000. Destroyers, cruisers, battleships, carriers, and many auxiliaries used the directors, with individual installations varying from one aboard destroyers to four on each battleship. The development of the Gun Directors Mark 33 and 37 provided the United States Fleet with good long range fire control against attacking planes. But while that had seemed the most pressing problem at the time the equipments were placed under development, it was but one part of the total problem of air defense. At close-in ranges the accuracy of the directors fell off sharply; even at intermediate ranges they left much to be desired. The weight and size of the equipments militated against rapid movement, making them difficult to shift from one target to another.Their efficiency was thus in inverse proportion to the proximity of danger.
The computer was completed as the Ford Mark 1 computer by 1935. Rate information for height changes enabled complete solution for aircraft targets moving over 400 miles per hour (640 km/h). Destroyers starting with the Sims class employed one of these computers, battleships up to four. The system's effectiveness against aircraft diminished as planes became faster, but toward the end of World War II upgrades were made to the Mark 37 System, and it was made compatible with the development of the VT (Variable Time) proximity fuze which exploded when it was near a target, rather than by timer or altitude, greatly increasing the probability that any one shell would destroy a target.

Mark 37 Director: The function of the Mark 37 Director, which resembles a gun mount with "ears" rather than guns, was to track the present position of the target in bearing, elevation, and range. To do this, it had optical sights (the rectangular windows or hatches on the front), an optical rangefinder (the tubes or ears sticking out each side), and later models, fire control radar antennas. The rectangular antenna is for the Mark 12 FC radar, and the parabolic antenna on the left ("orange peel") is for the Mark 22 FC radar. They were part of an upgrade to improve tracking of aircraft.
The director was manned by a crew of 6: Director Officer, Assistant Control Officer, Pointer, Trainer, Range Finder Operator and Radar Operator.
The Director Officer also had a slew sight used to quickly point the director towards a new target. Up to four Mark 37 Gun Fire Control Systems were installed on battleships. On a battleship, the director was protected by 1+1⁄2 inches (38 mm) of armor, and weighs 21 tons. The Mark 37 director aboard USS Joseph P. Kennedy, Jr. is protected with one-half inch (13 mm) of armor plate and weighs 16 tons.

Stabilizing signals from the Stable Element kept the optical sight telescopes, rangefinder, and radar antenna free from the effects of deck tilt. The signal that kept the rangefinder's axis horizontal was called "crosslevel"; elevation stabilization was called simply "level". Although the stable element was below decks in Plot, next to the Mark 1/1A computer, its internal gimbals followed director motion in bearing and elevation so that it provided level and crosslevel data directly. To do so, accurately, when the fire control system was initially installed, a surveyor, working in several stages, transferred the position of the gun director into Plot so the stable element's own internal mechanism was properly aligned to the director.
Although the rangefinder had significant mass and inertia, the crosslevel servo normally was only lightly loaded, because the rangefinder's own inertia kept it essentially horizontal; the servo's task was usually simply to ensure that the rangefinder and sight telescopes remained horizontal.
Mark 37 director train (bearing) and elevation drives were by D.C. motors fed from Amplidyne rotary power-amplifying generators. Although the train Amplidyne was rated at several kilowatts maximum output, its input signal came from a pair of 6L6 audio beam tetrode vacuum tubes (valves, in the U.K.).

Plotting room: In battleships, the Secondary Battery Plotting Rooms were down below the waterline and inside the armor belt. They contained four complete sets of the fire control equipment needed to aim and shoot at four targets. Each set included a Mark 1A computer, a Mark 6 Stable Element, FC radar controls and displays, parallax correctors, a switchboard, and people to operate it all.
(In the early 20th century, successive range and/or bearing readings were probably plotted either by hand or by the fire control devices (or both). Humans were very good data filters, able to plot a useful trend line given somewhat-inconsistent readings. As well, the Mark 8 Rangekeeper included a plotter. The distinctive name for the fire-control equipment room took root, and persisted even when there were no plotters.)

Ford Mark 1A Fire Control Computer: The Mark 1A Fire Control Computer was an electro-mechanical analog ballistic computer. Originally designated the Mark 1, design modifications were extensive enough to change it to "Mark 1A". The Mark 1A appeared post World War II and may have incorporated technology developed for the Bell Labs Mark 8, Fire Control Computer. Sailors would stand around a box measuring 62 by 38 by 45 inches (1.57 by 0.97 by 1.14 m). Even though built with extensive use of an aluminum alloy framework (including thick internal mechanism support plates) and computing mechanisms mostly made of aluminum alloy, it weighed as much as a car, about 3,125 pounds (1,417 kg), with the Star Shell Computer Mark 1 adding another 215 pounds (98 kg). It used 115 volts AC, 60 Hz, single phase, and typically a few amperes or even less. Under worst-case fault conditions, its synchros apparently could draw as much as 140 amperes, or 15,000 watts (about the same as 3 houses while using ovens). Almost all of the computer's inputs and outputs were by synchro torque transmitters and receivers.
Its function was to automatically aim the guns so that a fired projectile would collide with the target. This is the same function as the main battery's Mark 8 Rangekeeper used in the Mark 38 GFCS except that some of the targets the Mark 1A had to deal with also moved in elevation—and much faster. For a surface target, the Secondary Battery's Fire Control problem is the same as the Main Battery's with the same type inputs and outputs. The major difference between the two computers is their ballistics calculations.