It was all pretty crude technology and but it was rugged, simple, and easy to troubleshoot. And it all worked great for a long time until the Energy Star requirements came along and mandated lower energy use in washing machines and other appliances. Nowadays, the new front-loaders and the new high-efficiency top-loaders all use variable speed motors that require a special (and often expensive) phase control board. These types of motor arrangements are often called "variable-frequency drives" or "inverter drives." Wikipedia has a pretty good primer on this technology if you want to read more about it.
Anyway, back to the Old Skool motors. Why do we still care about these? Because there are still lots of them out there so any competent Appliantologist has to have a working understanding of both types of motor drives. Like the saying goes, "You can't figure out what's wrong if you don't know what 'right' is." IOW, how you have to understand how they're supposed to work so you can troubleshoot them when they're not working. Brother fairbank56 gives us a good, concise explanation of how these Old Skool motors operate...
For anyone interested in the theory. The motor in question is a single phase squirrel cage induction motor. A rotating magnetic field is required to get the rotor rotating. Single phase power does not provide this so we add another winding, the start winding. The way it is wound (number of turns and wire size) in conjunction with the series capacitor provides a phase shift in the current applied to it with respect to the current applied to the run winding. This creates a rotating magnetic field. Once the rotor is rotating, it creates its own rotating magnetic field and the start winding is no longer required and is switched out of circuit. On this particular motor, the start winding and capacitor are not designed for continuous power and will be damaged if not switched out of circuit after the motor starts or if it doesn't start. Starting direction of rotation is determined by the polarity of current through one winding with respect to the other winding. Reversing polarity of either winding will cause direction of rotation to reverse.
Source: Amana won't agitate in Regular Cycles