Non-Uniform Quantization
The idea behind nonuniform quantization (for example an a, or mu law for telephony speech) is to make better use of the input signal amplitude distribution in lossy source coding applications..
Over a period of time, it's emprically observed that, most of the speech occurs at low amplitude levels, and occasionally at high amplitude (full dynamic range) levels.
When this is the case, a uniform quantizer would be wasting its bits (its quantization levels) into many of those statistically less used louder signal levels.
In order to make efficient use of the quantizer (number of levels and bits), and to minimize the average quantization error (the distortion), it will be wiser to allocate more bits (finer levels) to the (often used) lower signal levels, while assigning less bits, and (coarse levels) to the (less used) louder signal levels.
The technical term for such a modification to a uniform quantizer is called a pdf optimized Max-Lloyd quantizer. The compander is a simplified example by assuming a fixed typical pdf of input speech and assigning a logarithmic non-uniformity to the step-sizes. This way, softer passages will be assigned more bits and finer resolution while louder passages will be assigned less (than uniform case) bits.
Futhermore, the distortion at the louder passages is perceptually less audible, due to psycho-acoustic principles, in particular to the masking effect; presence of a loud tone, suppresses the perception of nearby quiter tones, and makes them inaudible. Hence the quantization noise will be less audible while occuring with louder tones.