A musical note, a sensor reading, and a repeating animation may look complicated when drawn as one curve. Fourier's key idea is that many periodic signals can be described as a weighted sum of simple sine and cosine waves called harmonics.
From one wave to a sum
The fundamental frequency completes one cycle in the signal's period. Its harmonics complete two, three, or more cycles in that same period. A Fourier series assigns each harmonic a coefficient that says how much of that frequency is present, plus a constant term for the signal's average value.
For example, an ideal symmetric square wave can be approximated by adding odd sine harmonics: the fundamental, then the third harmonic at one-third amplitude, the fifth at one-fifth, and so on. Adding terms makes the flat sections and sharp transitions more recognizable.
Read the spectrum
A time-domain plot shows a signal changing over time. A frequency spectrum instead shows which frequencies contribute and how strongly. A single smooth sine wave has one dominant frequency; a sharper shape needs more high-frequency components. This view is useful in audio, communications, image processing, and scientific measurement.
Approximation has limits
A finite number of harmonics cannot reproduce a discontinuity perfectly. Near a sharp jump, the approximation overshoots; this is called the Gibbs phenomenon. More harmonics make the overshoot narrower, but it does not disappear entirely. Real digital signals are also sampled, so sampling rate and the Nyquist limit matter: frequencies above half the sample rate cannot be represented reliably.
Try changing the waveform and the number of harmonics below. Watch both the reconstructed curve and its spectrum; the controls make the relationship between a signal's shape and its frequency ingredients easier to see.