Distortion is what happens to a sound when a signal has a greater amplitude, or is louder, than the signal path can handle. The peaks of the wave get squished in order to fit, which changes the waveform and therefore the sound of the signal. It is kind of like pushing an 8 inch piece of paper through a 6 inch mail slot. It will fit, but the paper will be crinkled on the sides, never being the same again.
Distortion is commonly used in guitar amps and is found in small amounts in any analog circuit. Analog distortion is part of what makes people love old gear and tape machines. It can be described as character, color and warmth in these devices. Digital distortion is a whole other story. When a digital signal distorts there isn’t any pleasantness. There is just a squaring out of the wave, causing very unpleasant and audio ruining distortion.