How can I control an AC ceiling fan speed using Arduino Uno R3? I cannot understand the key behind controlling AC voltage using Arduino (though I know how to on/off ac appliances using arduino and relays).
Controlling the speed of an AC motor is not an easy undertaking, and messing around with mains voltages if you're not experienced and know what you're doing can be fatal.
Now that's the disclaimer out of the way, on to your question.
Unlike DC, you cannot use PWM to control the average power of AC. That is, after all, what PWM does - the on-off ratio creates a percentage of the power - but it can only be done if the incoming power is of a steady state.
For AC there are two ways of reducing the output power - either reduce the voltage, which can be done with resistors (look inside a multi-speed fan and you'll see resistors to change the speed); or chop up the AC waveform, which is a little like PWM but more specialized.
If you only want specific speeds then you could pre-define them using resistors and switch different resistor combinations in and out using relays. It's crude, but simple, and also probably the safest option for the beginner.
If you want proper variable speed, though, then you're going to have to chop up the waveform. The theory is quite simple: Start where the waveform crosses 0V in either direction, with the output on, and turn it off again at some percentage of the way through the cycle.
Like PWM it's the on-off percentage that defines the output power (and thus the speed), but timing is critical - you must start the PWM sequence at 0V and it must last exactly one cycle.
A number of simple circuits work in the opposite way - they remain off until the input waveform reaches a certain point, then switch on, until they reach the zero-crossing point. While simple, these circuits aren't good. The reason being that the sudden rise from 0V to the "trigger" voltage when the output becomes live can cause large spikes and noise in the circuit, which can cause lights to blow etc. It's the same effect as why bulbs most often blow when you turn them on, not when they have been running for a while - it's that very rapid rise from 0V to a high voltage which causes them to be over-stressed and blow.
So any circuit you choose should really be a proper one which turns on at 0V and turns off after a period, not turn on at a voltage and turn off at 0V.