The quantizing refers to each sample point. For example, lets say you have a 22khz recorded sample of something.
If you had an 8bit signal, each sample (22,000 are recorded per second) has a size of a single byte, and can record 256 different levels. So if you were sampling audio, for example, the different levels would correspond to amplitude of the signal at that time.
A 16bit signal stores a word for each sample, and so can have 65k different volume variations at a given point. Obviously, this gives a much better approximation to the actual sine-wave signal. If you know about Fourier analysis, you know that sharp edges in the time-domain correspond to high frequencies; so those rough and abrupt transitions in 8bit sound cause high-frequency noise.
So the total amount of size taken by an audio sample would be :
TIME of sampe x Sampling Rate x SampleSize
so for 3 minutes (180s) of audio recorded at 22khz at 8bit, you need 3.969MB of storage space, or double that if you either double the sampling rate, or double the sample size (8bit->16bit) Thus why compression is so handy. 🙂