Quantization is the process of converting a continuous range of values into a finite range of discreet values. This is a function of analog-to-digital converters, which create a series of digital values to represent the original analog signal. The bit depth (number of bits available) determines the accuracy and quality of the quantized value.
Quantization has a number of applications in digital image and audio production.
Digital audio involves taking an analog waveform (i.e. sound waves) and converting it to a series of individual samples, each of which has an amplitude value. The range of possible amplitude levels are defined by the bit depth, e.g. 8-bit quantization = 256 possible values; 16 bit quantization = 65,536 possible values, etc.
Quantization can also refer to the process of correcting the timing of a musical performance. The music track is analysed and stretched in time so that beats are evenly distributed, eliminating timing errors. Some manufacturers refer to quantizing features as autocorrect.
Video & Image Quantization
Digital image quantization is the process of determining which parts of an image can be discarded or consolidated with minimal subjective loss. Image quantization is inherently lossy however (i.e. the image quality is reduced due to the loss of some information).
Color quantization is a method of reducing the number of colors required to represent an image. For example, converting a photograph to GIF format requires the number of colors to be reduced to 256.
In images scanners, quantization is the process of forcing similar gray levels to the same level as a result of limited tonal resolution. Most often apparent in the shadow areas of scanned images.
There are many more definitions and uses for quantizing. Here are a few quick examples...
Values can be "rounded" to a commonly-agreed standard for simplicity. For example, a person's age is usually quantized to the number of years they have been alive as of their last birthday.
In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory.
In linguistics, a quantized expression is an expression that is applied to an entity but cannot be applied to any proper subparts of that entity. For example, "table" is a noun that can only apply to the whole table, not any individual part of the table. "Air" is a noun that can be applied equally to a large body of air or a smaller subpart of that air. "Table" is quantized, "air" is not.