I notice that the higher the number of poles, the slower the induction motor spins. Likewise, the lower the number of poles, the faster it spins. How does the number of poles affect its speed? Also, does the number of poles have any effect on its output torque?
During half of a cycle of the power frequency, the position of a rotor pole moves from one stator pole to the opposite pole of the same pair. If the motor has only two poles, that rotor pole moves back to the original position during the next half cycle. So there is one revolution per cycle of the power frequency. If there are more than two poles, the rotor moves from one pair of poles to another pair of poles during one cycle of the power frequency. Thus Speed(RPM) = 120 f(Hz) / P(poles).
In an induction motor, the mechanical speed of the rotor is less than the speed of the magnetic fields described above. The difference allows the stator to induce current in the rotor. The difference is very small when the motor has no external mechanical load. For the most common 3-phase motor designs, the slip is about 2 or 3 percent at full-load torque. For single-phase motors, the full-load slip is a little higher.
Power is speed multiplied by torque. Therefore a motor of a given power rating will have a higher torque rating if the speed rating is lower.
The physical size, weight and cost of a motor are more closely related to its torque rating than to power its power rating.
Not accounting for slippage, the rotor will move one pole-pair every cycle of the AC current. So the more poles you have, the more cycles it take to complete a full revolution.
In general, torque is a function of current. But it's more complex in an induction motor, as torque is also a function of the relative angle between the magnetic flux in the rotor and stator (slippage). There are a number of design parameters that determine the current and the slippage, and the number of poles only affect the torque to the extent that they effect those other parameters.
I had trouble imagining inductance in a 4+ pole motor. What helped me visualize the situation was to consider it the other way around. I.e. imagine a 4 pole generator which by definition has two bent (L shaped magnets) with their same polarities facing each other. This generator produces a full cycle sine wave per half rotation of its rotor. Here's a relevant youtube video (not mine). Therefore, in a 2N pole motor, you get 1/N speed of a 2 pole motor.