That's one big subject you're looking at!
Effectively what determines the ability of an engine to produce power from the chemical energy in the fuel is how much fuel it can take in, and how efficiently it can burn it.
There are two main ways of getting more fuel into a cylinder.
Firstly increase the volume, either by increasing the diameter of the bore or by increasing the stroke of the piston.
Secondly you can increase the ability to burn fuel by forcing more air into the cylinder, using a supercharger or turbocharger. The larger amount of oxygen available means you can then inject more fuel and get more power from the same volume.
(I'm not even going to go into other ways of getting more oxygen into the cylinder, such as nitrous injection, not really relevant to your average aero engine anyway)
So, if we choose to use a larger volume, we have several problems. First is weight. A larger cylinder and piston requires greater mass of metal to support it adequately. Of course, for a given volume, one large cylinder will usualy carry less of a weight penalty than several smaller ones, but that's a different story.
Some of the problems with a very large cylinder include (for an Avgas/mogas system) getting the charge to burn evenly, getting all the fuel to vapourise properly (ok I know, not ALL the fuel) and moving the fuel/air mixture in, and the burnt gases out, of the cylinder. Hence large cylinders tend to run at a much lower speed than smaller cylinders.
To get round the first of these, aero engines almost without exception use twin sparking systems. Hence the flame front starts from two points simultaneously and you get a more even burn. Hence the slightly rougher running you expect when switching to single mag systems during engine testing. In order to get the fuel/air mixture to work properly you need either a large bore carburettor or on more modern engines an injection system. With very large pistons, the speed of airflow required to ensure a proper mixture in the cylinder can get very high, and the manifold pressure (actually a partial vacuum for normally aspirated engines) can get very low. This brings increased risk of carb/inlet manifold icing and if the air supply cannot be maintained sufficiently, fuel starvation.
Even with most fuel injected systems (direct injection systems are slightly different, but not a great deal) you are dealing with very low pressures in the inlet manifold which, added to the heat absorbed by the fuel in vaporising, can leave you at severe risk of carb/manifold icing.
Of course most engines tend to use several smaller cylinders to provide the same swept volume as on larger cylinder. Now despite the weight penalty incurred in supporting structure, it is possible to get far higher power outputs from a many-cylindered engine than a single cylinder of similar volume. As I'm sure you can imagine, it reduces the vacuum in the inlet manifold, requires less of an air velocity to provide proper fuelling and can rotate much faster as each cylinder is much lighter than in the singl cylinder case.
Now in the case of turbo charging/supercharging (effectively the same thing, just the method of driving the compressor is different) you actually force the inlet manifold pressure up, so much more air can flow from the manifold into the cylinder while the inlet valves are open. This allows for much more power to be produced per cycle (four strokes as I'm sure you know, induction (suck) compression (squeeze) ignition (bang) and exhaust (blow)).
So back to the original question, what effects the power output of a cylinder? Firstly it's highly unusual to consider a single cylinder, as the combined effect of four, six, eight, ten, twelve, sixteen or even greater numbers of cylinders in the case of some of the realy large rotary/radial engines, is greater than the sum of the power-production ability of a single cylinder.
Now the power output of an individual engine is affected by the factors outlined above, as well as a few others such as the design/manufacture of the air intake system, the shape of the crown of the piston head, the shape of the cylinder top, the design of the exhaust system, the method of fuel supply, the type of fuel supplied..... the list goes on forever.
Hope some of that is helpful.