In digital audio, headroom is the space between your signal's peak level and 0 dBFS (the absolute maximum before digital clipping). Recording with adequate headroom — peaking at −12 to −6 dBFS — ensures you never distort even if the performer unexpectedly gets louder.
At 24-bit recording, you have 48 dB more dynamic range than 16-bit. This means you can comfortably record at lower levels (peaking at −18 dBFS) and still have a noise floor far below audibility. Many professional engineers record at conservative levels specifically to preserve headroom for mixing — they can always boost later without hearing noise, but they cannot undo digital clipping.
The concept applies to analog equipment too: analog headroom is the range between the nominal operating level and the point where the circuit begins to saturate or distort. Some analog gear is designed to be driven into this 'saturation zone' for musical effect (tape machines, tube preamps, analog compressors). Digital clipping is never musical.