The term "non-linear" is used in audio to describe a process where the output is not directly proportional to the input in a linear manner. Clipping is an example of a non-linear process.
"Linear" refers to a straight line. In a basic graph where the "X" axis represents input and the "Y" axis represents output, a straight line with a 45 degree slope passing through 0,0 would represent "unity gain" where the the output level matches the input level at every point.
A line with a different slope would represent some amount of "gain," with slopes higher than 45 degrees representing increased gain and below 45 degrees representing "loss." For example, a straight line that passes through 0,0 and 1,2 would represent a gain of 2. For any given input level, the output would be the input level multiplied by 2. Because this relationship is true for any input level, this is still considered to be a "linear" function.
One of the most basic forms of a non-linear function is manner in which human beings perceive differences in loudness level. The dB scale is logarithmic as versus linear. Most Volume controls are designed with a logarithmic characteristic for this reason; so as the Volume control is turned up or down, the volume of the audio will sound like it is changing at a constant rate. If a "linear" potentiometer is used as a Volume control, the perceived level of the audio will change very little at one end of the range and then very rapidly as the other end of the range is approached. This is due to the non-linear character of human hearing.
A non-linear process in audio is usually characterized as a form of “distortion.” Some examples are clipping, analog tape saturation, "hard limiting" with very fast attack and release times, or distortion caused by transformers or tubes. Another example is a bad galvanic connection (between metal contacts), which can sometimes act in a manner similar to a diode and only allow signals above a certain level to "pass" normally.
Non-linear processes can also take the form of digital emulation of analog circuitry such as emulation of tubes, transformers, or analog tape.