At the conclusion of Physics 24, you may have had the opportunity to play an instrument or sing while the sounds you made were sampled digitally and “Fourier transformed” to reveal the frequency components of the sounds. Sound waves are pressure variations that travel through a medium, such as the air. At any given point in space, the pressure varies as a function of time. But our ears do not perceive sounds with a time resolution of fractions of a millisecond. Rather, they perform a marvelous frequency decomposition so that we hear time-dependent tones (frequencies).
The idea of Fourier analysis is that we can represent any function of time in either the time domain or in the frequency domain; the Fourier transform allows us to go back and forth between the two representations of the “same function.”
If we are really thinking about a periodic signal, the analogous idea is a Fourier series, in which we represent the periodic signal in a series of sines and cosines whose fundamental frequency is determined by the period of the signal. While Fourier series are typically represented with the “real” functions sine and cosine, it is rare to use anything but the complex exponential representation for Fourier transforms.