the sine wave
The sine wave is the smooth, rolling up-and-down shape you get from a point going around a circle at steady speed, the most basic, pure wiggle in nature. A swinging pendulum, a plucked string's pure tone, and the mains voltage all trace this same shape.
A sine voltage is v(t) = Vp times sin(2 times pi times f times t), where Vp is the peak height, f is the frequency, and t is time. It has a single frequency, one pure pitch with no harmonics. Mathematically it is special: pass a sine through any resistor, capacitor, or inductor and you get back a sine of the same frequency, only scaled and shifted in time. No other shape keeps its form that way.
Because of that property, engineers analyse circuits one sine at a time and add the results. Any signal (a square wave, speech, music) can be broken into a sum of sines, called Fourier analysis, so understanding the sine unlocks everything else. Caveat: a perfectly pure sine is an ideal; real generators add a little distortion (extra harmonics), and a true single-frequency sine would have to last forever.
Pluck a tuning fork and the air pressure it makes traces a near-perfect sine, one clean pitch. Strike a real bell and you get several sines at once: the fundamental plus overtones.
A pure tone is a single sine.
Only a sine survives a linear circuit unchanged in shape. This is why we analyse with sines even when the real signal is a square wave or music.