Some quotes from 'Flying Guns – World War 1: Development of Aircraft Guns, Ammunition and Installations 1914-32' by Emmanuel Gustin and myself should give you the general idea:
"The first design for a device to synchronise the firing of the gun or guns with the rotation of the propeller was patented by the ingenious Franz Schneider, of whom we have already heard (although Blériot was also working in this field). However, his first patent in July 1913 was more precisely an interrupter rather than a synchronising gear; i.e., the mechanism prevented the gun from firing while a propeller blade was in front, instead of positively firing it when the line was clear. This latter approach was the method eventually adopted, although the term "interrupter gear" remained in popular but inaccurate use thereafter. It should be noted that a synchronising gear effectively turned the machine gun into a semi-automatic weapon, as it only fired one shot for each firing impulse received. It seems that Schneider had thought of this method as well, so should not be denied the credit. His patent envisaged a flexible synchronised gun, albeit with a limited range of movement.
This was followed in April 1914 by the Frenchman Raymond Saulnier's patent mechanism which used an oscillating rod to fire the gun (a flexible link was also proposed). More significantly, Saulnier also built the first practical synchronising gear at this time, but suffered from applying this to a Hotchkiss which was inherently unsuitable for synchronisation. Obviously discouraged by the results, Saulnier invented the steel deflector wedges as a simpler solution."
"Earlier systems had all used a mechanical linkage driven by a cam on the engine crankshaft (or in some cases the camshaft). The longer the distance from the gun to the propeller over which the linkages had to operate, the more problems there were with backlash, wear, heating and cooling. Very careful maintenance was required to keep them functioning properly, and malfunctions were common. The use in the CC gear of hydraulic pipes to transmit the pulses, or percussive wave transmissions, permitted more precise control and thereby the highest rate of fire, helped by the fact that two firing signals were sent per propeller revolution. It should be emphasised that this was not a conventional hydraulic system; the liquid in the pipes did not move, but merely transmitted the sonic pulses at very high speed, the ingenious invention of George Constantinesco. It was theoretically capable of sending 2,400 firing impulses per minute to a two-gun installation, except that no gun at that time was capable of firing at such a rate. The first aircraft tests were in August 1916 but it did not go into service until March 1917. There were significant teething problems, but the British persevered and the CC gear became their standard system until synchronised installations were abandoned in the Second World War."
"Synchronisation systems reduced the gun's natural rate of fire; by how much depending on a variety of factors. The first factor was a gun whose trigger mechanism could be controlled separately from the action of the bolt. As we have seen, the Lewis gun did not have this feature and the initial efforts at synchronisation resulted in a RoF of only 100-150 rpm, less than a quarter of the normal rate. The later Alkan gear managed to increase this to 160-200 rpm, and a more thorough redesign of the firing mechanism by Hazleton did see some limited use, but by then the Vickers had been accepted as standard. The second factor was a precise and reliable synchronising gear. The more accurate it was, the lower were the safety margins required and the greater the number of degrees of the propeller disk available for firing. The third factor was the gun's normal rate of fire; the higher this was, the greater the percentage loss through synchronisation (other things being equal). The final factor was the number of propeller blades; the more there were, the more critical accurate timing became.
The rate of fire of a synchronised gun tended to be rather erratic because it varied with propeller speed. Theoretically, it was possible to achieve an ideal match between the propeller rpm and gun's natural rate of fire, so that the gun was not slowed down at all. However, such harmony would obviously disappear as soon as the engine slowed down or speeded up. This particular problem was not solved until the adoption of the constant-speed propeller, which was uncommon until the late 1930s.
The effect of synchronisation on the rate of fire can best be explained by describing a simple system like that introduced by Fokker, in which one firing signal was sent to the gun for each rotation of the propeller. If the gun was capable of firing at 500 rounds per minute, then for propeller speeds of up to 500 revolutions per minute the RoF would be the same as the propeller rpm. However, as soon as the propeller exceeded 500 rpm, the gun mechanism could no longer keep up and could then only fire on every other rotation, so the RoF would drop to 250 rpm. It would then accelerate again with increasing propeller speed but at half the rate, so when the propeller was spinning at 1,000 rpm, the gun would be back to firing at 500 rpm again. Once more, propeller revs faster than this would cause the RoF to drop, but this time only to two-thirds of the full RoF, as it would fire on every third rotation, so it would be achieving 330 rpm. As the propeller continued to accelerate to 1,500 rpm, the gun would be back up to 500 rpm again, and so on. Any quoted figure for synchronised rates of fire could therefore only be an average. It is also worth repeating that quoted RoFs for unsynchronised guns were only averages also, with the actual RoF for different examples of the same type of gun varying quite significantly depending on age and maintenance. Any one gun might also vary in its rate of fire depending on the ammunition used, on the effect of the low temperatures experienced at high-altitude in congealing the gun lubricants and of course on the variable G-forces consequent on manoeuvring.
More advanced systems like the CC and the later German types sent two firing signals per propeller revolution (logical with a two-bladed propeller, in which there would be two firing opportunities per revolution), although possibly at the expense of some reliability in these primitive early systems, as they would have to work twice as fast. In this case, the maximum RoF for our 500 rpm gun would be reached twice as often, at 250, 500, 750, 1,000, 1,250 and 1,500 rpm. A still more sophisticated variation was to use a “critical sector cam”, which instead of just sending a single firing impulse sent a continuous one during the “safe” period when the propeller blades were out of the way. The effect of this was much less regular, with the gun firing in erratic bursts, but the average RoF was the highest of all."
Tony Williams: Military gun and ammunition website