Fourier series
A Fourier series says that any periodic signal, no matter how jagged, can be rebuilt by stacking up pure sine and cosine waves at multiples (harmonics) of its fundamental frequency. Picture a synthesizer: hold one note and add its 2nd, 3rd, 4th harmonics in just the right amounts and you can mimic a violin, a square wave, or a trumpet. Jean-Baptiste Fourier shocked mathematicians in 1807 by claiming even a sharp-cornered square wave is secretly a sum of smooth sinusoids.
The coefficients tell you 'how much' of each harmonic is present — this is the signal's spectrum, its recipe in the frequency domain. A square wave, for instance, contains only odd harmonics (1st, 3rd, 5th...) with amplitudes falling off as 1/n, which is why a buzzer sounds harsh and bright. Crucially, this is the gateway concept: it reveals that signals live equally well as a shape in time OR as a set of frequency ingredients, and switching between those two views is the heart of signal processing.
Fourier series is for periodic signals; when a signal isn't periodic, you stretch the idea into the Fourier transform, whose spectrum becomes continuous instead of a comb of discrete harmonics.