That’s not an easy question to answer, because I can think of about two dozen different AC motor designs of the top of my head most of which do not use capacitors. Since you are asking on this forum I’m guessing you want to know about motors found on most lathes, mills, band saws etc.…
Here goes:
No Cap motors are usually Split-Phase motors, these use a secondary winding that has a different size winding to create rotation. Split-Phase motors take longer to get up to speed and have less torque than capacitor motors. Usually found on small appliances that require low cost reliable motors. Please note that some older equipment has flat capacitors hidden under the motor frame. These can be dangerous because people don’t see the capacitor housing. They think the motor won’t shock them only to get a jolt and getting sliced up from an end mill or lathe tool.
Single Cap motors can be Start Capacitor motors or Run Capacitor motors. First off the Start Cap motor, these are essentially Split-Phase motors that use a capacitor to help the motor get up to speed faster. Once the motor reaches approx. 75-80% of its full rpm a centrifugal mechanical switch takes the capacitor out of the circuit. The Run Capacitor motor uses its capacitor to create a rotating magnetic field to create rotation. The capacitor on the Run Capacitor motor is energized as long as the motor is powered up.
Two Cap motors are almost always Start-Run Capacitor motors. These motors use a Start capacitor and a Run capacitor. Many times you will see two different size capacitor housings on the motor body. The Starter capacitor is used to increase the motors initial torque while the Run capacitor is used to increase the motors efficiency at rpm. They are more expensive to build but have higher torque needed for most single phase equipment. It’s extremely rare to see these motor sized greater than 7.5hp.