I have a fan in my house, It has an electronic PCB that responds to a remote control. The PCB has a voltage dropping capacitor so, I think it is supplied with a very low voltage such as 9 or 12 volts. There is no relays on the PCB nor inside the fan, How can the PCB control the 220 volts AC induction motor ?
A triac or alternistor is the normal way to control relatively high current AC (other than relays). Alternistors are similar to triacs, but behave better with inductive loads where regular triacs can fail to commutate (turn off). For example, the BTA12:
The triac can be used to switch windings (discrete speeds) or for phase control. Here is a Freescale app note showing a ceiling fan speed control design using phase control.
The transformer and opto-isolator shown are for safety reasons and could be omitted in a carefully designed and manufactured commercial product (to save money).