Clipping occurs when the peaks of an audio waveform get so large that they exceed your microphone or DAW's maximum capabilities. To deal with this issue, the waveform peaks are cut off (or "clipped"), turning a sine wave into a square wave. This results in nasty distortion, and though some guitarists intentionally overdrive their amps to create this effect, it doesn't sound pleasant in most other cases. So what is audio clipping, in a nutshell? For most musicians and producers, it's when your audio signal exceeds 0dBFS (decibel full scale) — the maximum level for mixing.
Digital vs. Analog Clipping
When clipping occurs with digital audio, the peaks are chopped off in a drastic way. However, today's digital audio workstations offer a newfangled approach to processing audio, with a technique called floating point processing. This gives you nearly unlimited headroom for your mix, as long as you keep the audio in a digital format. Routing that clipped digital audio into analog gear will cause problems if the signal is too loud, though. For those who are starting with an analog signal, clipping can create a nice softening effect if the peaks are not too extreme.
Results of Clipping
How to Prevent Clipping
The simplest way to avoid audio clipping is to lower your levels! However, if you're trying to boost a particular channel (i.e. a kick drum or snare) so that it fills the entire range, you can use a limiter to ensure that the signal doesn't exceed a certain threshold. When working in a digital recording environment, it's okay if the occasional channel goes into the red, but the master channel should not be clipping. Trust your ears — if a kick drum sounds better when clipped, then leave it the way it is.
Repairing Audio That is Already Clipped
It's much easier to take preventative measures when recording, rather than try to repair clipped audio after the face. In some cases, however, it's possible to recreate the segments of an audio signal that are lost due to clipping. Plugins like iZotope Rx3, Sony Sound Forge, and Audacity Clip Fix can be used to restore a damaged signal, and they all employ distinct mathematical methods to guess what information is missing.