Concerning the older, simple transformer, rectifier, capacitor AC to DC converters, what determines available amps (amp output)? Is it simply the thickness of the transformer wires?
What about the lighter, smaller, more complex, more modern converters. What factor/component determines available amps (amp output) for them?
Wall warts are designed to meet a certain balance of cost, size, efficiency, reliability, output quality, and output power. Each wall wart will have a different balance. Apple might focus on size and efficiency, while Samsung might care more about cost. The designer of the wall wart will understand the goals, the balance, that is required and design the wall wart accordingly.
There are too many designs to really go into the details about what makes one design have higher output power than another. Suffice it to say that EVERYTHING matters. Even the plastic case that the wall wart is inside matters. Just like if you are designing a passenger airplane to go the farthest distance, then even the seats matter. For a wall wart, the transformers, caps, diodes, wires, integrated circuits, inductors, resistors, MOSFETs, all matter. You can't just change the caps and expect the whole thing to suddenly output more power. You have to look at everything.
Let's use a car as an example. Let's say that you have a typical car that you want to make go faster. Of course you could replace the engine block to hold more and larger pistons. But then you need a bigger air intake so you can burn more fuel. And you need a bigger fuel pump. Now you need a stronger crankshaft, to handle the power. And a better transmission, because the old one would get destroyed under the stress. And a better driveshaft. And better axles. And better wheels/tires. And better brakes, so you can stop this. And better suspension to handle the increased torque. And better radiator to cool the hotter engine. Etc. The same is true for a wall wart-- but with electronic stuff instead of gear-head stuff.
Since you didn't specify a wall wart, I'm going to assume you mean a regulated wall wart. There is more going on inside there than a transformer. There is always rectification, filtering, and regulation. All of these things affect the current capacity of the wall wart.
Again, you didn't specify a wallwart, so I'm going to assume SMPS. Same deal as the linear regulation, only there is even more going on inside these wall warts. No single component inside these power supplies determines the current, assuming they are well designed. A poorly designed power supply will always have some kind of bottleneck.