Distortion in audio is very closely related to noise. Both "distortion" and "noise" are used to describe unwanted components of an audio stream, so what's the difference? I already defined noise when I talked about dynamic range. But what is distortion? The simplest answer is that distortion is signal-correlated artifacts. An artifact, as you remember, is anything unwanted in an audio signal. To be signal-correlated means that the volume of the artifact increases and decreases with the volume of the signal. To check, look at all of the artifacts in your signal. Now, increase your signal volume. Any artifact that increased is distortion. Anything that stayed constant is noise.
The most common form of distortion occurs when you have a signal (the fundamental) playing at frequency f, and you hear answering artifacts at frequencies 2f, 3f, 4f, and so on. This type of distortion is called harmonic distortion, and the artifact at frequency N*f is called the Nth harmonic. The distortion coming from a purely linear filter (In audio engineering terms, a filter is any device or process that takes a signal and outputs another signal, usually altered in some way) is harmonic. Harmonics also have some very interesting mathematical properties that I may describe in a future post.
Not all harmonic distortion is bad. It is harmonic distortion that gives a voice or instrument its timbre, which is nothing more than a characteristic signature of harmonics that is recognized by the ear to be different sounds. Different patterns of harmonic distortion are why a bass guitar, a cello, and a man's voice all sound different, even though they can all produce the same note.
Of course distortion is still undesirable in recording and playback equipment such as a computer, for the simple reason that the it changes a signal from what was intended. Non-harmonic distortion produces dischord. Harmonic distortion gives a signal a subtle and artificial timbre shift. A distorted voice may sound slightly garbled, or have a ringing to it. Distorted music might have notes that are just a bit off, or instruments that jangle. Finally, listening to a distorted signal causes listening fatigue, headaches, and tension that results in the urge to shut off that blasted noise. Your ear knows, even if your brain doesn't.
As an example, radio stations often broadcast distorted signals, as a result of overamplifying the antenna's output power. If you've ever been in the car and decided to shut off music that you like because you're just "tired of listening to it", then now you know why.