Short answers, pulled from the story.
A fire-control system is an integrated system of components that assists a gunner in achieving accurate firing through full or semi automation. It performs the same task as a human gunner but tries to do so faster and more accurately.
Arthur Pollen and Frederic Charles Dreyer independently developed the first naval fire-control systems. Pollen began work after seeing poor naval artillery accuracy at a gunnery practice near Malta in 1900, building on Lord Kelvin's proposal to use an analogue computer for the problem.
Radar incorporated into the fire-control system early in World War II let ships conduct effective gunfire at long range in poor weather and at night. Feeding radar data into the rangekeeper made long-range night engagements feasible for the first time.
At the Third Battle of Savo Island in November 1942, an American warship engaged Kirishima at a range of 8400 yards at night. Kirishima was set aflame, suffered explosions, and was scuttled by her crew after being hit by at least nine 16-inch rounds out of 75 fired, a 12% hit rate.
The Low Altitude Bombing System, or LABS, was a computerized bombing predictor integrated into nuclear-armed aircraft by the start of the Vietnam War. The computer itself gave the bomb release command at a calculated release point, allowing accurate release even while the aircraft maneuvered, and was originally designed for a tactic called toss bombing.
The radar-based M-9/SCR-584 system directed air defense artillery from 1943 and, combined with the VT proximity fuze, shot down V-1 cruise missiles with fewer than 100 shells per plane, compared to thousands in earlier systems. It was instrumental in the defense of London and Antwerp against the V-1.
Modern digital fire-control computers are installed on ships, submarines, aircraft, tanks, and even some small arms, such as the grenade launcher developed for the Fabrique Nationale F2000 bullpup assault rifle. They can aim machine guns, small cannons, guided missiles, rifles, grenades, and rockets, any weapon whose firing parameters can vary.