What is the noise whitening process?
Consider a WSS (wide sense stationary) discrete-time random process (RP) with i.i.d. samples $X[n,s]$ whose auto-correlation sequence (ACF function) is $\phi_{XX}[m]$. We call a random process as white-noise if its ACF is the following:

$$ \phi_{XX}[m] = \begin{cases} 
{\sigma_X^2 ~~~~~, \text{ for } m=0 \\ 0 ~~~~~~~~, \text{ otherwise} }\end{cases} $$ 

This also means that the RP is uncorrelated. If a RP does not have this property then it's a colored noise. And it means its samples are correlated.

When a RP is colored, it may be possible to find an LTI filter which will try to remove the correlation between the samples of the noise and hence make it whitened. Such a filter is called either as a decorrelator or equivalently as whitening filter.

It's best described in the context of AR-p (Auto-Regressive) random processes generation through all pole filtering, where the whitening filter will be the inverse of that all-pole AR filter which generates the colored noise from an input white noise.