Why the conversion to intermediate frequency?
This answer is focussing on radio receivers such as AM and FM.
If you are only interested in receiving a signal from one station you may not need to have or use an intermediate frequency. You can build your receiver to tune in to just that frequency - the tuning needs to be sharp - you need to reject all possible other sources that may pollute the signal you want.
This is done by a bunch of band pass filters that together, have a passband that is wide enough to cope with the signal you wish to receive but not so wide that it lets others in.
Now say you wanted to tune in to 2 stations - you'd have to re-align all this filtering to coincide with a new station. Historically radios were simple and moving a bunch of tuned band pass filters to a new centre frequency would be hard.
It was a lot easier to have a bunch of fixed band-pass filters that did the majority of all the unwanted channel rection rather than trying to align them as you tuned the dial.
Thus super-heterodyne receivers were conceived. The incoming broad range of many radio stations were "mixed" with an oscillator that can be simply tuned with a dial - this produced sum and difference frequencies and usually the difference frequency became the new "wanted" frequency. So for FM (88MHz to 108MHz), the I.F. frequency became 10.7MHz and the oscillator would be (typically) at 98.7MHz for tuning 88MHz signals and at 118.7MHz for tuning 108MHz signals.
Don't hang me on this - it could equally be at 77.3MHz rising to 97.3MHz to produce the same set of difference frequencies. Maybe someone can modify my answer or advise me on this.
It's a small matter though because the point is that once you were able to manipulate the incoming signal's carrier frequency you can feed the result through a tightly tuned fixed set of band-pass filters before you demodulate.
A bit more info about the VHF FM band
It goes from 88MHz to 108MHz and has an IF that is just slightly bigger (10.7MHz) than half the frequency range it covers. There is a sensible reason - if the oscillator were exactly tuned to pick up 88MHz (i.e. osc = 98.7MHz) the difference frequency it would produce from the top of the band at 108MHz would be 9.3MHz and this would be just out of band of the tuning centred at 10.7 MHz and therefore "rejected".
Of course if someone started transmitting just outside the FM band you may pick this up but I believe that legislation prevents this.
Following recent activity in this question I remembered that there is another valid reason for using an intermediate frequency. Consider that the signal from an antenna might be in the order of 1 uV RMS and then consider that you'll probably want the radio circuit to amplify this to something like 1V RMS (forgive the hand waving) at the demodulator. Well, that's a gain of 1 million or 120 dB and, no matter how hard you might try, having a circuit board with a gain of 120 dB is a recipe for feedback disaster i.e. it will oscillate and turn into a "theramin".
What an IF gets you is a break in the signal chain which prevents oscillation. So, you might have 60 dB of RF gain then convert to your IF and have 60 dB of IF gain - the signal at the end of the chain is no longer frequency compatible with what happens at the antenna and therefore, there is no theramin effect!
Some radios might have two intermediate frequencies - for just this reason alone you can reduce the RF gain to 40 dB and each IF stage can have a gain of 40 dB and NO theramin.
IF makes the receiver both more economical and higher quality. RF parts are trickier to make and use, and the circuitry more beset with problems of stray capacitance, inductance, noise, ground loops and interference. More so the higher the frequency. But we must have an RF front end because the signal at the antenna connection is just too weak to do anything with but amplify it. Necessary but expensive, designers want to minimize the amount of RF circuitry.
OTOH, we want good selectivity. Transmissions are allotted bandwidth, and multiple transmitters are under pressure to be squeezed together next to one another in frequency. We want a flat passband for the desired frequency, and complete blockage of frequencies outside that. Perfection is impossible but tradeoffs can be made for a "good enough" filter. This takes advanced filter design, not just a plain LC tuned circuit. While this could be done in RF, in theory, in practice it'll be tricky and expensive, and hard to make stable against temperature changes and aging.
We can make better filters meeting complex response requirements at lower frequencies, e.g. tens of MHZ, or sub-MHz. The lower the frequency, the easier it is to design a decent approximation to a rectangle response function filter. Turns out that making the down-converter - the local oscillator and mixer - is relatively easy and economical. Overall the system is most economical with minimal RF front end amplifiers, a down converter, and a beefy well-designed IF section doing all the fancy filtering.
The main lesson points are: * The higher the frequency, the more expensive and troublesome it is. * Elaborate filter requirements (anything beyond an elementary tuned circuit) is best done at lower frequencies
I find it interesting that this design strategy has held up over decades for many different systems utilizing wildly different technologies. Old vacuum tube radios looking like wooden furniture in the 1930s-1940s, transistor radios in the 1960s, tiny cell phones and bluetooth devices today, giant radio astronomy telescopes, spacecraft telemetry, and more.
Basically it's to allow the demodulation circuit to be made very sensitive with a narrow bandwidth.
If the demodulation circuit had to be wideband (say, able to work for any frequency from 88-108 MHz for FM), keeping a flat response across the entire frequency range would be difficult. Instead, the tuner is wideband and then beat (heterodyned) to a single intermediate frequency and sent to a very optimized demodulation circuit.