Generalized (non-specific) summary of motor types:
Alternating Current (AC). They are engineered to run off a certain AC voltage, which is alternating at a specific frequency (such as 60 cycles per second). Most (but not all), can be sped up by increasing the frequency of the AC, or slowed down by lowering the frequency (up to a point). As frequency goes up, torque is typically reduced, as frequency goes down, the current drawn goes UP. Go too low and the coils go into "saturation" and they get VERY hot and draw massive current. You will see people call the boxes which do this a "VFD".
Synchronous AC motors. Other than the clock on your wall, there is not a significant use for these types of motor in a shop. I won't go into them further.
Direct Current (DC). These motors run on a smooth flow of electricity which does not alternate. An example of DC is the power out of a car battery. The starter in your car is a DC motor, as is your windshield wiper motor and the blower motor for the air conditioning in your car. All three of these DC motors have "brushes" which makes contact with the armature. Brushes do wear with time, and with enough use, require replacement. Typically, as applied voltage goes up, the RPMs and Power produced by Brushed DC motors goes up. Too much voltage will burn up the coils in the motor.
Brushless DC Motors (BLDC). These motors do not use contacts on the armature to "switch" power to the magnetic coils in the motor. Instead, an EXTERNAL speed controller has connections to each coil in the motor, and applies power to each coil as it is needed to produce the desired rate of rotation. Or, to put it in other words, the DC power is switch by the external controller off and on in a sequential fashion to the coils, the faster it switches, the faster the motor spins (up to a limit). Some BLDC motor controllers sense RPM and LOAD, and will attempt to compensate how they drive the motor to keep the RPMs constant, as the load changes.