For most of the war, this was done using a device called a 'predictor' or 'director'. These were electromechanical computers that could calculate the path of an incoming aircraft. This could then be passed, through cables, to the guns. Here, a time fuse was set according to the calculations of the director; this could be done manually or using another device called a 'fuse setter', depending on the gun. Later in the war, the Allies introduced a 'proximity fuse'. This used a miniaturised radar in the nose of the shell to detect an enemy aircraft nearby. This meant that the shell could be detonated at exactly the right moment, and therefore was much more accurate and effective.
The Royal Navy's main AA director during WWII was the High Angle Control System, or HACS. To engage a target aircraft with HACS, the ship's Air Defence Officer on the bridge would designate a target with an air defence sight. This cued in the Director Control Tower (DCT) to the target's location. The DCT would track the target, measuring data about it. It mounted an optical rangefinder to measure the range to the target. Later models added radar systems to make a more accurate measurement of range. The aircraft's course could be determined using its angle relative to the ship. The speed of the target was estimated by the personnel in the DCT. These data were passed to the Transmitting Station, which housed the ship's fire control computers. Here, a 'High Angle Calculating Table' would crunch the numbers, and determine the correct direction to point the guns and the fuse settings. These were then sent to the guns. The gun crews would read off the azimuth, elevation and fuse setting from dials which tracked the HACS results. The guns were then trained and elevated and the fuses set manually. HACS was a fairly good system, but had several major flaws. As originally designed, it was intended to track and engage a single aircraft; it could not fire a barrage at a given altitude and position. This made it hard to use against dive-bombers in particular, though there were work-arounds. During the war, a barrage unit was developed and added, which greatly simplified the use of this essential tactic. HACS was also less accurate than its equivalents in other navies, the USN in particular. HACS relied on estimates of the target aircraft's speed, and assumed that the target's speed course and altitude remained constant. Other navies systems were 'tachymetric' - they relied on actual measurements of the target aircraft's speed. This made them more accurate than the British system, if the other inputs were comparably accurate.
Before 1940, the flak shells used timed fuses. Firing the weapon caused the shell to begin to count down, and then the aim and distance were calibrated so that the shell would explode at a certain altitude corresponding to the timing of the shell. It was the same system for artillery, with the goal being for the shell to be timed to explode in the air above a ground target for maximum dispersal of shrapnel at optimal speed.
In 1940, the National Defense Research Committee of the United States asked for what was impossible at the time: a proximity fuse. They asked the Carnegie foundation and Johns Hopkins University to get to work on it, and they did. One of the famous researchers on the project was Dr. James Van Allen, better know for his Van Allen radiation belt in astronomy.
They had a monumental task: to shrink down a radio transmitter and receiver into the space of a shell fuse. The technology didn't exist in 1940. But they did it, and they got a shell that worked about half the time. That seems lousy now, but in WW2 it would sometimes require hundreds of shells fired to even come close to bringing down an aircraft. A 50% success rate was a dream. The new weapon (called the VT fuse) was first deployed at Guadalcanal, where two measly shells (out of the normal 50+) of this new type from the USS Helena brought down a Japanese dive bomber. The shell is even purported to have "saved" Van Allen himself, on USS Washington at the Battle of the Philippine Sea.
The new proximity shells were not authorized for use as artillery over land until 1944, due to the very sensitive nature of the technology. It was (rightfully) feared that allowing the Germans to find a dud proximity shell would be disastrous. By 1944 the Allies had decided that the benefits outweighed the risks. The proximity shells played a major role in the Battle of the Bulge, and Patton loved them.'
Here is a nice writeup.
Here is another with a technical schematic.
This is also an interesting document from 1946.