Learn the basics: www.phy.davidson.edu/instrumentation/NEETS.htm
There are uncountable variations of design. Usually you have to use one or the other type of motor because only one type of power is available. If you need variable speed, that would be a DC motor. If you need a precise speed, that would be a synchronous AC motor. There are endless compromises.
Dot matrix printers used stepper motors to move the paper. They were driven by DC to move the motor one precise step at a time by switching power to the different coils. Hobbyists took the motors out and drove them with AC at carefully controlled frequency to get a precise speed. Here is an example: www.altmann.haan.de/turntable/
The most common AC motor is the shaded pole. It is cheap and simple and the speed is fairly reliable. Most of the different designs are simply different methods of producing the needed phase shift to get a rotating magnetic field.
Thanks for the awesome description. I really appreciate it. I think it would be useful for me to make sure I am comfortable with being able to explain the differences between ac and dc all together. That would make for a great starting point.