Picture two identical twins walking down a noisy street side by side. Every car horn, every shout, every burst of static hits them both equally. Now imagine you only care about the difference between what the twins say, not what they each individually heard. Subtract one twin's experience from the other's and all that shared street noise cancels out, leaving only the part where they actually disagreed. That's common-mode rejection in a sentence: noise that arrives equally on both signal paths gets thrown away, and only the difference (the part that actually matters) survives.
How it actually works
A balanced cable sends your audio signal twice through two separate wires (called hot and cold). The cold copy is deliberately flipped in polarity before it travels. Any interference the cable picks up along its run (electrical hum, radio noise, whatever) gets induced into both wires equally and in the same direction, since they're sitting right next to each other. At the receiving end, a differencing circuit flips the cold signal back and subtracts it from the hot signal. The original audio, which was intentionally inverted, comes back out doubled and intact. The noise, which was never inverted in the first place, cancels itself out in the subtraction.
The technical term for "noise that hits both wires equally" is common-mode noise (mode here just means "the way the signal is riding on the wires," and common means it's identical on both). Common-mode rejection is just the act of getting rid of it.
Why it matters for DJs
This is the actual physics behind the advice "use balanced cables for long runs." It's not superstition or gear-snobbery: a balanced connection has a built-in noise-cancellation system, and an unbalanced one doesn't. The longer the cable, or the closer it runs past lighting rigs and power cables, the more common-mode noise it picks up, and the more that rejection capability actually earns its keep.
In the context of Headroom
Common-mode rejection comes up in Module 2: Signal Flow right alongside balanced vs. unbalanced wiring and ground loops. Once you understand the mechanism, "balanced cables are quieter" stops being a rule you follow and becomes a rule you understand.