Fourier Transform in real life
Your title indicated you were interested in the use of FT's in real-life.  One application where I have made use of FTs is the decoding of touch-tone signals, like those generated on a telephone keypad. Each "touch-tone" is a superposition of two tones, selected from a set of four columns (1209 Hz, 1336 Hz, 1477 Hz, 1633 Hz) and four rows (697 Hz, 770 Hz, 852 Hz, 941 Hz).  (Wait, you say your telephone only has 3 columns of buttons?  There is actually a provision for four more buttons along the right side labelled A, B, C and D, which was used in the military.)

This picture shows what the combination of 941 Hz and 1336 Hz looks like on an oscilloscope:



One can see there is a regular pattern here, but no way to separate out the two tones.

If however you run this signal into a spectrum analyzer, which performs a FFT (Fast Fourier Transform), you get this:



The two peaks correspond to the frequencies 950 Hz and 1330 Hz.  Not exactly the ones listed earlier, but this probably represents slightly-off oscillators in the telephone keypad.  The frequencies chosen for the eight tones were designed so the two frequencies in each pair would be far enough apart to allow for some leeway.

Back before microcontrollers were invented (and even after, when they were not fast enough to handle this sort of thing), decoding of touch-tones was done in hardware using a set of analog filters, one for each of the eight frequencies.  Here is a Western Electric touch-tone decoder, circa 1965.  The analog filter circuitry is contained on the plug-in cards to the right (with the orange markings).



This circuitry was later reduced to an IC, first the LM567 (one chip per column) and then the HT9170 which used highly accurate switched capacitor filters to decode all 16 combinations of tones.

With microcontrollers, and particularly DSPs (Digital Signal Processors), things have become much easier.  Initially DSPs were separate chips, and had to be combined with a microprocessor to form a complete system.  Now they can be combined into one chip.

I used a Microchip dsPIC33FJ256GP710 (which, as the name implies contains a DSP in addition to a general purpose microcontroller based on the PIC24, and costs around $6). I also used a free DTMF decoder library (most of it written in assembly language) that performs the FFT, and generates two outputs corresponding to the peaks in the earlier picture.  So I was able to decode DTMF tones coming into the microcontroller's ADC without adding any addition hardware.  Meanwhile the microcontroller was free to perform all of the other functions of the device I was building.