I want to add some things to this thread as well:
Load is what the generator is attached to. A light bulb, a space heater, a motor etc.
Brush DC motors can be used as a generator with no modifications.
It uses a permanent magnet and a commutator to make the motor move. DC is the same kind of voltage that you see in a battery.
AC generators generate AC currert or current that has a sinusoidal output. Generally the magnetic field is changed by varying the stator current. Thus stator cuurent regulates voltage. RPM of the engine regulates frequency. Residual in the stator is designed such that the AC generator can start. The lack of permanent magnets reduces weight.
Alternators, which are now commonly used in cars, use the same technique, they generate 3 phase AC at a frequency dependent on engine rotation. All the phases are rectified and turned into DC. This answers you question concerning AC generators charging batteries.
In some portable generators, both AC and DC voltages are created with different windings.
Three phase motors allow rotation because of the relationship of the sine wave voltages for each phase. They are also more efficient than the induction motor.
A special case of the DC motor is a n phase brushless motor. N is usually >=3. There are sensors that sense the angular position of the motor, such that it can be determined when to energize each particular winding.
Each winding is generally supplied with a pulse width modulated signal that determines speed.
To see how this might work with a DC brushed motor. If one applied the full voltage at 10% of the time, The motor will run much slower, but would essentially have the same torque as the higher speed.
So, DC motors can be used as generators, motors and in vehicle applications when they are used as a generator they create dynamic braking.
AC generators are generally rated in kVA. DC generators in Watts. The reason why is VA, is because of the phase relationship and different voltage and current waveforms.
If the voltage and cuurent are sinusoidal and in phase then ohms law works.
If the voltages and currents are out of phase then the power formula is replaced by P= V*I*Cos(theta). Cos(0) =1 for resistive loads.
If the VI relationships are non-linear and not in phase, but periodic, you need true RMS meters to measure the voltage and currents.
If they are not periodic then Power is difficult to measure.
The magnetic field question is a physics question.
Design of a generator is probably more involved because transformer design is an involved art too. There is eddy current losses, copper losses, thermal losses, wire spacing losses etc.
The basic equations escape me, but I think they are in a physics text somewhere.
Again with the basic equations, try the physics forum. Maybe a question phrased as:
What are the governing equations that affect DC generator design. Research what you know.
I found this:
DC Generator Theory Summary