Here's how a radio works in seiyafan's own words:
In the earlier days communication often involves transmitting and receiving a limited number of signals. Later as the number of signals increases, however a problem is getting obvious. For cars on a highway, if you want more cars to go through certain point one of the common ways is to increase the number of lanes. In other words, some cars need to be driven in parallel rather than in series. In communication this idea is extended to the extreme, if there are 30 radio stations in your area, 30 different signals need to be sent out simultaneously without interfering with each other.
Signal modulation was first introduced as a solution to this problem. What modulation does it that the original signal, say Pavarotti's singing, is multiplied by a carrier signal. Then this modulated signal becomes the one that gets transmitted through a medium, it could be a piece of wire or air. At the receiving end, the signal gets demodulated back to its original form. This is the basic of AM radio.
At the transmitting end each signal from different stations is multiplied by a different carrier signal. So if you are listening to 880kHz or 1010kHz, that is the frequency of the carrier signal sending out by that radio station. And each station broadcasts at a frequency that is at least 10kHz more or less than one of the other station, this way it leaves enough room to transmit the voice signal. For AM the bandwidth is about 5kHz, for FM it's higher, that's the reason why AM has lower fidelity than FM. CD has higher quality than FM because the bandwidth of CD covers the entire human hearing range, so no information is filtered out. That's also where the term Hi-Fi originates, high-fidelity.
Now back to the station example, so now you got 30 signals all being transmitted at the same time, and Pavarotti's singing is mixed with the other 29. The antenna in your radio will pick up all the signals, so now the question is if you just want to hear Pavarotti’s singing, what do you do?
In the time domain the signal picked up by your radio is gonna look messy because it's a mixer of everything, including noise. However if you display the signal in the frequency domain, things are getting clear because each station transmits at its own frequency. If you want to hear Pavarotti's singing, you just tune your radio to that station, say 920kHz. It's not hard to do if you can build a tuner circuit and an amplifer to amplify the signal. But what is really going on inside?
When a signal is modulated, its frequency is changed. So in order to demodulate the signal, we have to change the frequency back. This is done by multiplying the modulated signal with another carrier signal that’s exactly the same as the one used for modulation. The reason is that a multiplication in the time domain equals to a convolution in the frequency domain. Since the carrier signal only occurs at one frequency, it is just a step impulse, aka the delta function by Fourier transform. By the sampling property, convolution of a signal with delta function is just a shift of the signal to that frequency. This is how signal is shifted in modulation, and also how it’s shifted back in demodulation. Then when we apply a filter to get rid of everything but the shifted signal, we get Pavarotti’s singing.
But this has a problem. In order for this theory to work, we have to design the demodulator such that the carrier signal it generates has to exactly the same as that generated by the radio station. Simple math shows that if the two differs by a mere Pi/2, you will get zero as your output, therefore making the tuner very sensitive to small changes. Another problem is that the filter needs to be changed with the tuner and good, dynamic filter is difficult to build. 100 years ago people didn't want to design complicated tuner. So then what happened?
In the 1910's, Armstrong solve this problem using his superheterodyne receiver. The key idea here is that instead of trying to match the carrier signal, the modulated signal is shifted to a higher frequency. So now the carrier signal always occurs at a fixed frequency (455 kHz is what's used then and also what's used today) above the modulated frequency. When this carrier signal gets multiplied with the modulated signal, the result will be a signal always present at 455 kHz. Since we can design really good filters at fixed cutoff frequencies, we will get the original signal back but shifted to a higher frequency. Now the problem is to restore the signal back to its original and this is very easy to solve. You can build a half-way rectifier circuit as a so-called envelope detector. And then just feed the signal to your power amp and your speaker, and there's your favorite Pavarotti's singing.
