How to control the speed of an induction motor?

The simplest means to adjust the speed of a (squirrel cage) induction motor is to vary the load applied. Since the rotor is not separately powered, adding load will reduce the shaft speed (i.e. slow the motor down). Removing load will increase the shaft speed. The upper limit is the slip speed (which is slightly below synchronous speed) for the applied frequency; the lower limit is zero speed or "stall" condition.

This does not mean the load adjustment is the best answer in terms of efficiency (of the motor, or of the process), or in terms of general system performance.

All AC machines have a "synchronous" speed determined by the relationship between the applied frequency and the number of magnetic poles created by the windings. More poles results in slower shaft speeds for the same frequency.

The actual equation is
120 * (frequency in Hertz) / (number of poles) = (shaft speed in rpm)

For squirrel cage induction machines, this is complicated by the fact that the rotor always turns slightly slower than the frequency applied to the stator winding (i.e. "slips").

This means that to vary the rotational speed of a motor requires a modification to either the applied frequency OR to the number of poles created by the winding. Since the motor is already designed and built, changing the poles is not really an option. Therefore, changing the frequency becomes the only really viable alternative.

Using power electronics - such as a pulse-width modulated (PWM) drive or other variable-frequency topology will effectively enable control of the frequency seen by the motor independent from the actual supply frequency from the utility. The result is an "adjustable" speed of operation. Although the computing power of modern variable frequency drives is usually sufficient to approximate shaft speed and position from current and flux calculations, the precision of the control can be improved by having hardwired feedback of rotor speed / position, generated from a physical sensor that maybe of either a contact or non-contact type. This is especially true at very low operating speeds where there is a relatively small change in frequency between operating points.

How efficient the new "drive + motor" system will be is at least partially dependent on the desired operating parameters: if it's a single speed point, the motor by itself is more efficient (i.e no additional drive losses).