Just a very minor nit, because some days it seems that God has led me to Stunthanger for the express purpose of disagreeing with Brett Buck over totally insignificant issues (but, this is an
engineering group, and disagreeing over nits is one of the things that engineers do):
The "other kind" of motor commonly found in power tools isn't synchronous. They're called induction motors and they're almost, but not quite, synchronous -- this is why you see speeds that are almost, but not quite, 3600 RPM (60Hz), or 1800 RPM (60Hz/2, because it's a multi-pole winding), or (more rarely) 1200 RPM. Here's the skinny:
https://en.wikipedia.org/wiki/Induction_motor.
However, both Brett and Phil are entirely correct that an induction motor is going to be very unhappy running on reduced voltage: it'll be trying it's best to run at full speed, which means using it's usual amount of electrical power. But the voltage will be lower, which means it'll have to suck more current, which means that it'll get hot and, in the worst case, it'll let out the magic smoke and cease to function (and stink up your shop). Phil is also correct that a VFD will solve that problem, although I've got zero experience with it.
There
are such things as synchronous motors. Old electro-mechanical clocks use a kind of self-starting synchronous motor called a hysteresis motor. Synchronous motors were used in mills and factories for bits that needed to be constant speed and that didn't have to have controlled speed. Such motors might have power ratings up to the hundreds of horsepower, and been as large as a car. The big synchronous motors weren't self-starting: they either had a pilot induction motor to get them going, or they were designed so that they would operate either as an induction machine for starting, or as a synchronous machine for running. Getting them running involved switching them over to being an induction machine, waiting until they were up to speed, then throwing a honkin' big switch (and probably ducking -- the closest I got to these things was in engineering lab in school, and even a 1HP synchronous machine makes some pretty alarming noises when you switch it over).
Just for grins, consider the fact that a brushless DC motor is essentially a synchronous machine with a permanent magnet providing the field (which means it can't be run as an induction motor). Instead of starting on induction and switching to synchronous operation, however, the ESC synchronizes itself to the motor, driving at a frequency that's tailored to the need of the machine it's turning. Brushless DC motors with Hall-effect commutation, similar to RC car motors, are making their way into high-end power tools for the same reasons of weight and battery economy that we like using them in our planes.