On a generator, you have a prime mover (say, an engine) connected to the actual generator, which consists of either rotating coils of wire within a magnetic field, or rotating magnets surrounded by coils of wire.
The number of poles (magnetic poles) and the rotational speed determine the output frequency: Freq = Engine_RPM * Number_Of_Poles / 120.
Typically, a United States portable generator runs at 3600 RPM, with 2 poles, for a design frequency of 60Hz. Larger portable generators run at 1800 RPM with 4 poles here.
That is how frequency is determined. The number of turns and the magnetic structure determine how many volts are produced at the design frequency, voltage and frequency aren't related in any fashion except for design. Again, in the States, most portable generators are wound to have a 240VAC single phase output, which is center tapped and delivered as two 120VAC hots with one neutral, but virtually any voltage can be delivered.
The current output of a generator is determined by its load, as long as the load doesn't exceed the maximum capacity of the generator's prime mover (engine) plus the conversion losses of the actual generator. Prime mover power is often rated in horsepower (US) or kilowatts (everywhere else). With no losses, a 10 horsepower engine could deliver 7457 watts (actually VA for non-resistive loads) continuously, or 62.1 amps at 120VAC continuously. Try to take more, and the engine will slow down (reducing both the frequency and the voltage, which will also drop the current) until you reach a point that the engine actually stalls.
You get fluctuation of frequency and voltage as the load changes because the engine cannot respond immediately to the actual load change. There are regulators controlling the engine throttle that attempt to keep the engine at a fixed (design) speed, but it takes time for the engine to respond to new commands as it has to deal with varying fuel/air mixtures and combustion which aren't instantaneous.
As a clarification to other discussions here:
For a purely resistive load, halving the voltage would halve the current, and result in one quarter the power consumed. You can't say that just cutting the voltage in half cuts the power consumed in half. With some devices, that may be true, but it entirely depends on the load.
Best Answer
Generators, transformers, motors, all have the same relationship for the maximum power of a winding.
The amount of power you can safely put through a winding depends on the amount of copper you use, measured either as mass or volume. It doesn't matter whether it's a few turns of thick wire, or many turns of thin wire, the current carrying capacity varies in inverse proportion to its working voltage, to yield constant power.
You can do a simple thought experiment with a winding space that has two identical coils of N turns wound on it. Put them in series for 2N turns of thin wire, put them in parallel for N turns of wire with twice the area. The same volts/amps on each individual coil adds up to the same total power handling for the same power dissipation within the coil, while the voltage and current at the terminals change by a factor of 2 depending on the connection.
While the voltage rating is quite easy to measure or compute, the current carrying capacity is more of a 'ratings' exercise. You have to do two things, (a) keep the windings cool enough and (b) keep voltage drops due to \$V=IR\$ within your specification.
For a short duty cycle, you can run higher currents than if you want to run continuously. As a rule of thumb, small to medium power transformers and motors tend to run about 3 A/mm² current density, but that can change substantially with different cooling arrangements or different interpretations of what small and medium mean. Fortunately it's quite easy to measure winding temperature when assessing how much current you can use over how much time. The resistance of copper increases by about 10% for every 25°C rise in temperature. Measure the winding resistance at room temperature, and then again after a period of running. I tend not to like exceeding 50°C rise, but then I'm a cautious soul.